Created the Real Analysis Theorems and Definitions packet

.gitignore

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+## Core latex/pdflatex auxiliary files:
+*.aux
+*.lof
+*.log
+*.lot
+*.fls
+*.out
+*.toc
+*.fmt
+*.fot
+*.cb
+*.cb2
+.*.lb
+
+## Intermediate documents:
+*.dvi
+*.xdv
+*-converted-to.*
+# these rules might exclude image files for figures etc.
+# *.ps
+# *.eps
+# *.pdf
+
+## Bibliography auxiliary files (bibtex/biblatex/biber):
+*.bbl
+*.bcf
+*.blg
+*-blx.aux
+*-blx.bib
+*.run.xml
+
+## Build tool auxiliary files:
+*.fdb_latexmk
+*.synctex
+*.synctex(busy)
+*.synctex.gz
+*.synctex.gz(busy)
+*.pdfsync
+
+## Build tool directories for auxiliary files
+# latexrun
+latex.out/
+
+## Auxiliary and intermediate files from other packages:
+# algorithms
+*.alg
+*.loa
+
+# achemso
+acs-*.bib
+
+# amsthm
+*.thm
+
+# beamer
+*.nav
+*.pre
+*.snm
+*.vrb
+
+# changes
+*.soc
+
+# comment
+*.cut
+
+# cprotect
+*.cpt
+
+# elsarticle (documentclass of Elsevier journals)
+*.spl
+
+# endnotes
+*.ent
+
+# fixme
+*.lox
+
+# feynmf/feynmp
+*.mf
+*.mp
+*.t[1-9]
+*.t[1-9][0-9]
+*.tfm
+
+#(r)(e)ledmac/(r)(e)ledpar
+*.end
+*.?end
+*.[1-9]
+*.[1-9][0-9]
+*.[1-9][0-9][0-9]
+*.[1-9]R
+*.[1-9][0-9]R
+*.[1-9][0-9][0-9]R
+*.eledsec[1-9]
+*.eledsec[1-9]R
+*.eledsec[1-9][0-9]
+*.eledsec[1-9][0-9]R
+*.eledsec[1-9][0-9][0-9]
+*.eledsec[1-9][0-9][0-9]R
+
+# glossaries
+*.acn
+*.acr
+*.glg
+*.glo
+*.gls
+*.glsdefs
+*.lzo
+*.lzs
+*.slg
+*.slo
+*.sls
+
+# uncomment this for glossaries-extra (will ignore makeindex's style files!)
+# *.ist
+
+# gnuplot
+*.gnuplot
+*.table
+
+# gnuplottex
+*-gnuplottex-*
+
+# gregoriotex
+*.gaux
+*.glog
+*.gtex
+
+# htlatex
+*.4ct
+*.4tc
+*.idv
+*.lg
+*.trc
+*.xref
+
+# hyperref
+*.brf
+
+# knitr
+*-concordance.tex
+# TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files
+# *.tikz
+*-tikzDictionary
+
+# listings
+*.lol
+
+# luatexja-ruby
+*.ltjruby
+
+# makeidx
+*.idx
+*.ilg
+*.ind
+
+# minitoc
+*.maf
+*.mlf
+*.mlt
+*.mtc[0-9]*
+*.slf[0-9]*
+*.slt[0-9]*
+*.stc[0-9]*
+
+# minted
+_minted*
+*.pyg
+
+# morewrites
+*.mw
+
+# newpax
+*.newpax
+
+# nomencl
+*.nlg
+*.nlo
+*.nls
+
+# pax
+*.pax
+
+# pdfpcnotes
+*.pdfpc
+
+# sagetex
+*.sagetex.sage
+*.sagetex.py
+*.sagetex.scmd
+
+# scrwfile
+*.wrt
+
+# svg
+svg-inkscape/
+
+# sympy
+*.sout
+*.sympy
+sympy-plots-for-*.tex/
+
+# pdfcomment
+*.upa
+*.upb
+
+# pythontex
+*.pytxcode
+pythontex-files-*/
+
+# tcolorbox
+*.listing
+
+# thmtools
+*.loe
+
+# TikZ & PGF
+*.dpth
+*.md5
+*.auxlock
+
+# titletoc
+*.ptc
+
+# todonotes
+*.tdo
+
+# vhistory
+*.hst
+*.ver
+
+# easy-todo
+*.lod
+
+# xcolor
+*.xcp
+
+# xmpincl
+*.xmpi
+
+# xindy
+*.xdy
+
+# xypic precompiled matrices and outlines
+*.xyc
+*.xyd
+
+# endfloat
+*.ttt
+*.fff
+
+# Latexian
+TSWLatexianTemp*
+
+## Editors:
+# WinEdt
+*.bak
+*.sav
+
+# Texpad
+.texpadtmp
+
+# LyX
+*.lyx~
+
+# Kile
+*.backup
+
+# gummi
+.*.swp
+
+# KBibTeX
+*~[0-9]*
+
+# TeXnicCenter
+*.tps
+
+# auto folder when using emacs and auctex
+./auto/*
+*.el
+
+# expex forward references with \gathertags
+*-tags.tex
+
+# standalone packages
+*.sta
+
+# Makeindex log files
+*.lpz
+
+# xwatermark package
+*.xwm
+
+# REVTeX puts footnotes in the bibliography by default, unless the nofootinbib
+# option is specified. Footnotes are the stored in a file with suffix Notes.bib.
+# Uncomment the next line to have this generated file ignored.
+#*Notes.bib
+
+# Draw.io backup files
+*.bkp
+*.dtmp
+

README.md

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+# Real Analysis Theorems and Definitions
+
+This repository houses LaTeX code (and the corresponding generated [PDF](./Real-Analysis-Theorems-and-Definitions.pdf))
+that generates a packet of all the theorems and definitions contained within
+[Introduction to Real Analysis, 4th Edition, by Robert G. Bartle and Donald R. Sherbert](https://a.co/d/3xSYuQf).
+
+## Purpose
+The purpose of this packet is to provide a useful "cheat sheet" for viewing all of the theorems and definitions within the
+textbook for easy reference when writing proofs.
+
+## What's Included?
+This packet includes all of the following from the textbook:
+
+* Definitions
+* Theorems
+* Corollaries
+* Lemmas
+* Remarks
+* Select examples that contain a definition or theorem
+* Occasional asides as they appear in the textbook

Real-Analysis-Theorems-and-Definitions.tex

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+\documentclass[12pt,letterpaper]{report}
+\usepackage{init}
+
+\newcommand{\dist}{\text{dist}}
+\author{Alexander J. Clarke}
+\title{Real Analysis Theorems and Definitions}
+\begin{document}
+\maketitle
+\clearpage
+\begin{center}
+	\thispagestyle{empty}
+	\vspace*{\fill}
+	All theorems, corollaries, lemmas, remarks, and asides are direct quotes from\\
+
+	\textit{Introduction to Real Analysis, 4th Edition, by Robert G. Bartle and Donald R. Sherbert}
+	\vspace*{\fill}
+\end{center}
+\tableofcontents
+
+\import{chapter-1/}{chapter-1.tex}
+\import{chapter-2/}{chapter-2.tex}
+\import{chapter-3/}{chapter-3.tex}
+\import{chapter-4/}{chapter-4.tex}
+\import{chapter-5/}{chapter-5.tex}
+\import{chapter-6/}{chapter-6.tex}
+\import{chapter-7/}{chapter-7.tex}
+\import{chapter-8/}{chapter-8.tex}
+\import{chapter-9/}{chapter-9.tex}
+\import{chapter-10/}{chapter-10.tex}
+\import{chapter-11/}{chapter-11.tex}
+
+\end{document}

chapter-1/chapter-1.tex

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+\chapter{Preliminaries}
+\subimport{./}{sets-and-functions.tex}
+\subimport{./}{mathematical-induction.tex}
+\subimport{./}{finite-and-infinite-sets.tex}

chapter-1/finite-and-infinite-sets.tex

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+\section{Finite and Infinite Sets}
+
+\begin{definition}
+	\begin{enumerate}
+		\item[]
+		\item The empty set $\emptyset$ is said to have $0$ \textbf{elements}.
+
+		\item If $n \in \N$, a set $S$ is said to have $n$ \textbf{elements} if there exists a bijection from the set $\N_n := \{1, 2, \dots, n\}$ onto $S$.
+
+		\item A set $S$ is said to be \textbf{finite} if it is either empty or it has $n$ elements for some $n \in \N$.
+
+		\item A set $S$ is said to be \textbf{infinite} if it is not finite.
+	\end{enumerate}
+\end{definition}
+
+\begin{theorem}[\textbf{Uniqueness Theorem}]
+	If $S$ is a finite set, then the number of elements is $S$ is a unique number in $\N$.
+\end{theorem}
+
+\begin{theorem}
+	The set $\N$ of natural numbers is an infinite set.
+\end{theorem}
+
+\begin{theorem}
+	\begin{enumerate}
+		\item[]
+		\item If $A$ is a set with $m$ elements and $B$ is a set with $n$ elements and if $A \cap B = \emptyset$, then $A \cup B$ has $m +n$ elements.
+
+		\item If $A$ is a set with $m \in \N$ elements and $C \subseteq A$ is a set with $1$ element, then $A \setminus C$ is a set with $m-1$ elements.
+
+		\item If $C$ is an infinite set and $B$ is a finite set, then $C \setminus B$ is an infinite set.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	Suppose that $S$ and $T$ are sets and that $T \subseteq S$.
+	\begin{enumerate}
+		\item If $S$ is a finite set, then $T$ is a finite set.
+
+		\item If $T$ is an infinite set, then $S$ is an infinite set.
+	\end{enumerate}
+\end{theorem}
+
+\begin{definition}
+	\begin{enumerate}
+		\item[]
+		\item A set $S$ is said to be \textbf{denumerable} (or \textbf{countably infinite}) if there exists a bijection of $\N$ onto $S$.
+
+		\item A set $S$ is said to be \textbf{countable} if it is either finite or denumerable.
+
+		\item A set $S$ is said to be \textbf{uncountable} if it is not countable.
+	\end{enumerate}
+\end{definition}
+
+\begin{theorem}
+	The set $\N \times \N$ is denumerable.
+\end{theorem}
+
+\begin{theorem}
+	Suppose that $S$ and $T$ are sets and that $T \subseteq S$.
+	\begin{enumerate}
+		\item If $S$ is a countable set, then $T$ is a countable set.
+
+		\item If $T$ is an uncountable set, then $S$ is an uncountable set.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	The following statements are equivalent:
+	\begin{enumerate}
+		\item $S$ is a countable set.
+
+		\item There exists a surjection of $\N$ onto $S$.
+
+		\item There exists an injection of $S$ into $\N$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	The set $\Q$ of all rational numbers is denumerable.
+\end{theorem}
+
+\begin{theorem}
+	If $A_m$ is a countable set for each $m \in \N$, then the union $A:= \bigcup\limits_{m=1}^{\infty} A_m$ is countable.
+\end{theorem}
+
+\begin{theorem}[\textbf{Cantor's Theorem}]
+	If $A$ is any set, then there is no surjection of $A$ onto the set $\mathcal{P}(A)$ of all subsets of $A$.
+\end{theorem}

chapter-1/mathematical-induction.tex

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+\section{Mathematical Induction}
+
+\begin{theorem}[\textbf{Well-Ordering Property of $\N$}]
+	Every nonempty subset of $\N$ has a least element.
+\end{theorem}
+A more detailed statement of this property is as follows: If $S$ is a subset of $\N$ and if $S \neq \emptyset$, then there exists $m \in S$ such that $m \leq k$ for all $k \in S$.
+
+\begin{theorem}[\textbf{Principle of Mathematical Induction}]
+	Let $S$ be a subset of $\N$ that possesses the two properties:
+	\begin{enumerate}
+		\item The number $1 \in S$.
+		\item For every $k \in \N$, if $k \in S$, then $k + 1 \in S$.
+	\end{enumerate}
+	Then we have $S = \N$.
+\end{theorem}
+
+\begin{theorem}[\textbf{Principle of Mathematical Induction (second version)}]
+	Let $n_0 \in \N$ and let $P(n)$ be a statement for each natural number $n \geq n_0$. Suppose that:
+	\begin{enumerate}
+		\item The statement $P(n_0)$ is true.
+
+		\item For all $k \geq n_0$, the truth of $P(k)$ implies the truth of $P(k+1)$.
+	\end{enumerate}
+	Then $P(n)$ is true for all $n \geq n_0$.
+\end{theorem}
+
+\begin{theorem}[\textbf{Principle of Strong Induction}]
+	Let $S$ be a subset of $\N$ such that
+	\begin{enumerate}
+		\item $1 \in S$.
+
+		\item For every $k \in \N$, if $\{1, 2, \dots \} \subseteq S$, then $k+1 \in S$.
+	\end{enumerate}
+	Then $S = \N$.
+\end{theorem}

chapter-1/sets-and-functions.tex

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+\section{Sets and Functions}
+
+\begin{definition}
+	Two sets $A$ and $B$ are said to be \textbf{equal,} and we write $A=B$ if they contain the same elements.
+	\\\\Thus, to prove that the sets $A$ and $B$ are equal, we must show that
+	\[A \subseteq B \text{ and } B \subseteq A\]
+\end{definition}
+
+\begin{definition}
+	\begin{enumerate}
+		\item[]
+		\item The \textbf{union} of sets $A$ and $B$ is the set
+		      \[A \cup B := \{x:x \in A \text{ or } x \in B\}.\]
+
+		\item The \textbf{intersection} of the sets $A$ and $B$ is the set
+		      \[A \cap B := \{x:x \in A \text{ and } x \in B \}.\]
+
+		\item The \textbf{complement of $B$ relative to $A$} is the set
+		      \[A \setminus B := \{x:x \in A \text{ and } x \notin B\}\]
+	\end{enumerate}
+\end{definition}
+
+\begin{theorem}
+	If $A$, $B$, $C$ are sets, then
+	\begin{enumerate}
+		\item $A \setminus (B \cup C) = (A \setminus B) \cap (A \setminus C),$
+		\item $A \setminus (B \cap C) = (A \setminus B) \cup (A \setminus C).$
+	\end{enumerate}
+\end{theorem}
+
+\begin{definition}
+	If $A$ and $B$ are nonempty sets, then the \textbf{Cartesian product} $A \times B$ of $A$ and $B$ is the set of all ordered pairs $(a,b)$ with $a \in A$ and $b \in B$. That is,
+	\[A \times B := \{(a,b): a \in A,\ b \in B\}.\]
+\end{definition}
+
+\begin{definition}
+	Let $A$ and $B$ be sets. Then a \textbf{function} from $A$ to $B$ is a set $f$ of ordered pairs in $A \times B$ such that for each $a \in A$ there exists a unique $B \in B$ with $(a,b) \in f$. (In other words, if $(a,b) \in f$ and $(a, b') \in f$, then $b = b'$.)
+\end{definition}
+
+\begin{definition}
+	If $E$ is a subset of $A$, then the \textbf{direct image} of $E$ under $f$ is the subset $f(E)$ of $B$ given by
+	\[f(E):=\{f(x):x \in E\}\]
+
+	If $H$ is a subset of $B$, then the \textbf{inverse image} of $H$ under $f$ is the subset $f^{-1}(H)$ of $A$ given by
+	\[f^{-1}(H):=\{x \in A:f(x) \in H\}\]
+\end{definition}
+
+\begin{definition}
+	Let $f:A \rightarrow B$ be a function from $A$ to $B$.
+	\begin{enumerate}
+		\item The function $f$ is said to be \textbf{injective} (or to be \textbf{one-one}) if whenever $x_1 \neq x_2$, then $f(x_1) \neq f(x_2)$. If $f$ is an injective function, we also say that $f$ is an \textbf{injection}.
+
+		\item The function $f$ is said to be \textbf{surjective} (or to map $A$ \textbf{onto} $B$) if $f(A)=B$; that is, if the range $R(f)=B$. If $f$ is a surjective function, we also say that $f$ is a \textbf{surjection}.
+
+		\item If $f$ is both injective and surjective, then $f$ is said to be \textbf{bijective}. If $f$ is bijective, we also say that $f$ is a \textbf{bijection}.
+	\end{enumerate}
+\end{definition}
+
+\begin{definition}
+	If $f: A \rightarrow B$ is a bijection of $A$ onto $B$, then
+	\[g := \{(b,a) \in B \times A: (a,b) \in f\}\]
+	is a function on $B$ into $A$. This function is called the \textbf{inverse function} of $f$, and is denoted by $f^{-1}$. The function $f^{-1}$ is also called the \textbf{inverse} of $f$.
+	\\We can also express the connection between $f$ and its inverse $f^{-1}$ by noting that $D(f)=R(f^{-1})$ and $R(f)=D(f^{-1})$ and that
+	\[b=f(a) \text{ if and only if } a=f^{-1}(b)\]
+\end{definition}
+
+\begin{definition}
+	If $f: A \rightarrow B$ and $g:B \rightarrow C$, and if $R(f) \subseteq D(g) = B$, then the \textbf{composite function} $g \circ f$ (note the order!) is the function from $A$ into $C$ defined by
+	\[(g \circ f)(x) := g(f(x)) \text{ for all } x \in A\]
+\end{definition}
+
+\begin{theorem}
+	Let $f: A \rightarrow B$ and $g: B \rightarrow C$ be functions and let $H$ be a subset of $C$. Then we have
+	\[(g \circ f)^{-1}(H) = f^{-1}(g^{-1}(H)).\]
+
+	Note the reversal in the order of the functions.
+\end{theorem}

chapter-10/chapter-10.tex

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+\chapter{The Generalized Riemann Integral}
+\subimport{./}{definition-and-main-properties.tex}
+\subimport{./}{improper-and-lebesgue-integrals.tex}
+\subimport{./}{infinite-intervals.tex}
+\subimport{./}{convergence-theorems.tex}

chapter-10/convergence-theorems.tex

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+\section{Convergence Theorems}
+
+\begin{theorem}[\textbf{Uniform Convergence Theorem}]
+	Let $(f_k)$ be a sequence in $\mathcal{R}^*[a,b]$ and suppose that $(f_k)$ converges \textbf{uniformly} on $[a,b]$ to $f$. Then $f \in \mathcal{R}^*[a,b]$ and
+	\[\displaystyle\int_{a}^{b}f=\lim\limits_{k \to \infty}\displaystyle\int_{a}^{b}f_k\]
+	holds.
+\end{theorem}
+
+\begin{definition}
+	A sequence $(f_k)$ in $\mathcal{R}^*(I)$ is said to be \textbf{equi-integrable} if for every $\varepsilon>0$ there exists a gauge $\delta_\varepsilon$ on $I$ such that if $\dot{\mathcal{P}}$ is any $\delta_\varepsilon$-fine partition of $I$ and $k\in\N$, then $\left|S(f_k;\dot{\mathcal{P}})-\displaystyle\int_If_k\right|<\varepsilon$.
+\end{definition}
+
+\begin{theorem}[\textbf{Equi-integrability Theorem}]
+	If $(f_k) \in \mathcal{R}^*(I)$ is equi-integrable on $I$ and if $f(x)=\lim f_k(x)$ for all $x \in I$, then $f \in \mathcal{R}^*(I)$ and
+	\[\displaystyle\int_If=\lim\limits_{k \to \infty}\displaystyle\int_If_k\]
+\end{theorem}
+
+\begin{definition}
+	We say that a sequence of functions on an interval $I \subseteq \R$ is \textbf{monotone increasing} if it satisfies $f_1(x) \leq f_2(x) \leq \dots \leq f_k(x) \leq f_{k+1}(x) \leq \dots$ for all $k \in \N$, $x \in I$. It is said to be \textbf{monotone decreasing} if it satisfies the opposite string of inequalities, and to be \textbf{monotone} if it is either monotone increasing or decreasing.
+\end{definition}
+
+\begin{theorem}[\textbf{Monotone Convergence Theorem}]
+	Let $(f_k)$ be a monotone sequence of functions in $\mathcal{R}^*(I)$ such that $f(x)=\lim f_k(x)$ almost everywhere on $I$. Then $f \in \mathcal{R}^*(I)$ if and only if the sequence of integrals $\left(\int_I f_k\right)$ is bounded in $\R$, in which case
+	\[\int_I f = \lim\limits_{k \to \infty} \int_I f_k.\]
+\end{theorem}
+
+\begin{theorem}[\textbf{Dominated Convergence Theorem}]
+	Let $(f_n)$ be a sequence in $\mathcal{R}^*(I)$ and let $f(x)=\lim f_k(x)$ almost everywhere on $I$. If there exist functions $\alpha, \omega$ in $\mathcal{R}^*(I)$ such that
+	\[\alpha(x)\leq f_k(x)\leq\omega(x)\ \ \text{  for almost every  }\ \ x \in I\]
+	then $f \in \mathcal{R}^*(I)$ and
+	\[\displaystyle\int_If=\lim\limits_{k \to \infty}\displaystyle\int_I f_k.\]
+	Moreover, if $\alpha$ and $\omega$ belong to $\mathcal{L}(I)$, then $f_k$ and $f$ belong to $\mathcal{L}(I)$ and
+	\[||f_k-f||=\displaystyle\int_I|f_k-f|\to 0\]
+\end{theorem}
+
+\begin{definition}
+	A function $f:[a,b]\to\R$ is said to be \textbf{(Lebesgue) measurable} if there exists a sequence $(s_k)$ of step functions on $[a,b]$ such that
+	\[f(x)=\lim\limits_{k \to \infty} s_k(x)\ \ \text{  for almost every  }\ \ x \in [a,b]\]
+	We denote the collection of measurable functions on $[a,b]$ by $\mathcal{M}[a,b]$.
+\end{definition}
+
+\begin{theorem}
+	Let $f$ and $g$ belong to $\mathcal{M}[a,b]$ and let $c \in \R$.
+	\begin{enumerate}
+		\item Then the functions $cf, |f|,f+g,f-g,$ and $f\cdot g$ also belong to $\mathcal{M}[a,b]$.
+		\item If $\varphi:\R \to \R$ is continuous, then the composition $\varphi \circ f \in \mathcal{M}[a,b]$.
+		\item If $(f_n)$ is a sequence in $\mathcal{M}[a,b]$ and $f(x)=\lim f_n(x)$ almost everywhere on $I$, then $f \in \mathcal{M}[a,b]$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	A function $f:[a,b]\to\R$ is in $\mathcal{M}[a,b]$ if and only if there exists a sequence $(g_k)$ of continuous functions such that
+	\[f(x)=\lim\limits_{k \to \infty} g(x)\ \ \text{  for almost every  }\ \ x \in [a,b]\]
+\end{theorem}
+
+\begin{theorem}[\textbf{Measurability Theorem}]
+	If $f \in \mathcal{R}^*[a,b]$, then $f \in \mathcal{M}[a,b]$
+\end{theorem}
+
+\begin{theorem}[\textbf{Integrability Theorem}]
+	Let $f\in\mathcal{M}[a,b]$. Then $f \in \mathcal{R}^*[a,b]$ if and only if there exist functions $\alpha, \omega \in \mathcal{R}^*[a,b]$ such that
+	\[\alpha(x)\leq f(x)\leq \omega(x)\ \ \text{  for almost every  }\ \ x \in [a,b]\]
+	Moreover, if either $\alpha$ or $\omega$ belongs to $\mathcal{L}[a,b]$, then $f \in \mathcal{L}[a,b]$.
+\end{theorem}

chapter-10/definition-and-main-properties.tex

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+\section{Definition and Main Properties}
+
+In \textit{Definition 5.2.2}, we defined a \textbf{gauge} on $[a,b]$ to be a strictly positive function $\delta:[a,b] \to (0,\infty)$. Further, a tagged partition $\dot{\mathcal{P}}:=\{(I_i,t_i)\}_{i=1}^n$ of $[a,b]$, where $I_i:=[x_{i-1},x_i]$, is said to be \textbf{$\delta$-fine} in the case
+\[t_i \in I_i \subseteq [t_i-\delta(t_i),t_i+\delta(t_i)]\ \ \text{for}\ \ i=1,\dots,n\]
+
+\begin{definition}
+	A function $f:[a,b] \to \R$ is said to be \textbf{\textit{generalized} Riemann integrable} on $[a,b]$ if there exists a number $L\in\R$ such that for every $\varepsilon>0$ there exists a gauge $\delta_\varepsilon$ on $[a,b]$ such that if $\dot{\mathcal{P}}$ is any $\delta_\varepsilon$-fine partition of $[a,b]$, then
+	\[|S(f;\dot{\mathcal{P}})-L|<\varepsilon\]
+	The collection of all generalized Riemann integrable functions will usually be denoted by $\mathcal{R}^*[a,b]$.
+	\\\\	It will be shown that if $f \in \mathcal{R}^*[a,b]$, then the number $L$ is uniquely determined; it will be called the \textbf{generalized Riemann integral of $f$} over $[a,b]$. It will also be shown that if $f \in \mathcal{R}[a,b]$, then $f \in \mathcal{R}^*[a,b]$ and the value of the two integrals is the same. Therefore, it will not cause any ambiguity if we also denote the generalized Riemann integral of $f \in \mathcal{R}^*[a,b]$ by the symbols
+	\[\displaystyle\int_{a}^{b}f\ \ \text{  or  }\ \ \displaystyle\int_{a}^{b}f(x)dx\]
+\end{definition}
+
+\begin{theorem}[\textbf{Uniqueness Theorem}]
+	If $f \in \mathcal{R}^*[a,b]$, then the value of the integral is uniquely determined.
+\end{theorem}
+
+\begin{theorem}[\textbf{Consistency Theorem}]
+	If $f \in \mathcal{R}[a,b]$ with integral $L$, then also $f \in \mathcal{R}^*[a,b]$ with integral $L$.
+\end{theorem}
+
+\begin{theorem}
+	Suppose that $f$ and $g$ are in $\mathcal{R}^*[a,b]$. Then:
+	\begin{enumerate}
+		\item If $k \in \R$, the function $kf$ is in $\mathcal{R}^*[a,b]$ and
+		      \[\displaystyle\int_{a}^{b}kf=k\displaystyle\int_{a}^{b}f\]
+
+		\item The function $f+g$ is in $\mathcal{R}^*[a,b]$ and
+		      \[\displaystyle\int_{a}^{b}(f+g)=\displaystyle\int_{a}^{b}f+\displaystyle\int_{a}^{b}g\]
+
+		\item If $f(x)\leq g(x)$ for all $x \in [a,b]$, then
+		      \[\displaystyle\int_{a}^{b}f \leq \displaystyle\int_{a}^{b}g\]
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}[\textbf{Cauchy Criterion}]
+	A function $f:[a,b] \to \R$ belongs to $\mathcal{R}^*[a,b]$ if and only if for every $\varepsilon >0$ there exists a gauge $\eta_\varepsilon$ on $[a,b]$ such that if $\dot{\mathcal{P}}$ and $\dot{\mathcal{Q}}$ are any partitions of $[a,b]$ that are $\eta_\varepsilon$-fine, then
+	\[|S(f;\dot{\mathcal{P}})-S(f;\dot{\mathcal{Q}})|<\varepsilon\]
+\end{theorem}
+
+\begin{theorem}[\textbf{Squeeze Theorem}]
+	Let $f:[a,b] \to \R$. Then $f \in \mathcal{R}^*[a,b]$ if and only if for every $\varepsilon>0$ there exist functions $\alpha_\varepsilon$ and $\omega_\varepsilon$ in $\mathcal{R}^*[a,b]$ with
+	\[\alpha_\varepsilon(x) \leq f(x) \leq \omega_\varepsilon(x)\ \forall\ x \in [a,b]\]
+	and such that
+	\[\displaystyle\int_{a}^{b}(\omega_\varepsilon-\alpha_\varepsilon) \leq \varepsilon\]
+\end{theorem}
+
+\begin{theorem}[\textbf{Additivity Theorem}]
+	Let $f:[a,b] \to \R$ and let $c \in (a,b)$. Then $f \in \mathcal{R}^*[a,b]$ if and only if its restrictions to $[a,c]$ and $[c,b]$ are both generalized Riemann integrable. In this case
+	\[\displaystyle\int_{a}^{b}f=\displaystyle\int_{a}^{c}f+\displaystyle\int_{c}^{b}f\]
+\end{theorem}
+
+\begin{theorem}[\textbf{The Fundamental Theorem of Calculus (First Form)}]
+	Suppose there exists a \textbf{countable} set $E$ in $[a,b]$, and functions $f,F:[a,b] \to \R$ such that:
+	\begin{enumerate}
+		\item $F$ is continuous on $[a,b]$.
+		\item $F'(x)=f(x)$ for all $x \in [a,b]\setminus E$.
+		      \\Then $f$ belongs to $\mathcal{R}^*[a,b]$ and
+		      \[\displaystyle\int_{a}^{b}f=F(b)-F(a)\]
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}[\textbf{Fundamental Theorem of Calculus (Second Form)}]
+	Let $f$ belong to $\mathcal{R}^*[a,b]$ and let $F$ be the indefinite integral of $f$. Then we have:
+	\begin{enumerate}
+		\item $F$ is continuous on $[a,b]$.
+		\item There exists a null set $Z$ such that if $x \in [a,b]\setminus Z$, then $F$ is differentiable at $x$ and $F'(x)=f(x)$.
+		\item If $f$ is continuous at $c \in [a,b]$, then $F'(c)=f(c)$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}[\textbf{Substitution Theorem}]
+	\begin{enumerate}
+		\item[]
+		\item Let $I:=[a,b]$ and $J:=[\alpha, \beta]$, and let $F:I \to \R$ and $\varphi:J \to \R$ be continuous functions with $\varphi(J)\subseteq I$.
+		\item Suppose there exist sets $E_f \subset I$ and $E_\varphi\subset J$ such that $f(x)=F'(x)$ for $x \in I\setminus E_f$, that $\varphi'(t)$ exists for $t \in J\setminus E_\varphi$, and that $E:=\varphi^{-1}(E_f)\cup E_\varphi$ is countable.
+		\item Set $f(x):=0$ for $x \in E_f$ and $\varphi'(t):=0$ for $t \in E_\varphi$. We conclude that $f \in \mathcal{R}^*(\varphi(J))$, that $(f \circ \varphi)\cdot \varphi^t \in \mathcal{R}^*(J)$ and that
+		      \[\displaystyle\int_{\alpha}^{\beta}(f \circ \varphi)\cdot\varphi^t=F\circ\varphi\left.\right|_\alpha^\beta=\displaystyle\int_{\varphi(\alpha)}^{\varphi(\beta)}f\]
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}[\textbf{Multiplication Theorem}]
+	If $f \in \mathcal{R}^*[a,b]$ and if $g$ is a monotone function on $[a,b]$, then the product $f \cdot g$ belongs to $\mathcal{R}^*[a,b]$.
+\end{theorem}
+
+\begin{theorem}[\textbf{Integration by Parts Theorem}]
+	Let $F$ and $G$ be differentiable on $[a,b]$. Then $F'G$ belongs to $\mathcal{R}^*[a,b]$ if and only if $FG'$ belongs to $\mathcal{R}^*[a,b]$. In this case we have
+	\[\displaystyle\int_{a}^{b}F'G=FG\left.\right|_a^b-\displaystyle\int_{a}^{b}FG'\]
+\end{theorem}
+
+\begin{theorem}[\textbf{Taylor's Theorem}]
+	Suppose that $f,f',f'',\dots,f^{(n)}$ and $f^{(n+1)}$ exist on $[a,b]$. Then we have
+	\[f(b)=f(a)+\frac{f'(a)}{1!}(b-a)+\dots+\frac{f^{(n)}(a)}{n!}(b-a)^n+R_n\]
+	where the remainder is given by
+	\[R_n=\frac{1}{n!}\displaystyle\int_{a}^{b}f^{(n+1)}(t)\cdot(b-t)^n dt\]
+\end{theorem}

chapter-10/improper-and-lebesgue-integrals.tex

@@ -0,0 +1,72 @@
+\section{Improper and Lebesgue Integrals}
+
+\begin{theorem}[\textbf{Hake's Theorem}]
+	If $f:[a,b] \to \R$, then $f \in \mathcal{R}^*[a,b]$ if and only if for every $\gamma \in (a,b)$ the restriction of $f$ to $[a,\gamma]$ belongs to $\mathcal{R}^*[a,\gamma]$ and
+	\[\lim\limits_{\gamma \to b^-}\displaystyle\int_{a}^{\gamma}f=A\in\R\]
+	In this case $\displaystyle\int_{a}^{b}f=A$.
+\end{theorem}
+
+\begin{definition}
+	A function $f \in \mathcal{R}^*[a,b]$ such that $|f| \in \mathcal{R}^*[a,b]$ is said to be \textbf{Lebesgue integrable} on $[a,b]$. The collection of all Lebesgue integrable functions on $[a,b]$ is denoted by $\mathcal{L}[a,b]$.
+\end{definition}
+
+\begin{theorem}[\textbf{Comparison Test}]
+	If $f,\omega \in \mathcal{R}^*[a,b]$ and $|f(x)| \leq \omega(x)$ for all $x \in [a,b]$, then $f \in \mathcal{L}[a,b]$ and
+	\[\abs{\displaystyle\int_{a}^{b}f}\leq\displaystyle\int_{a}^{b}|f| \leq\displaystyle\int_{a}^{b}\omega\]
+\end{theorem}
+
+\begin{theorem}
+	If $f,g \in \mathcal{L}[a,b]$ and if $c \in \R$, then $cf$ and $f+g$ also belong to $\mathcal{L}[a,b]$. Moreover
+	\[\displaystyle\int_{a}^{b}cf=c\displaystyle\int_{a}^{b}f\ \ \text{  and  }\ \ \displaystyle\int_{a}^{b}|f+g|\leq\displaystyle\int_{a}^{b}|f|+\displaystyle\int_{a}^{b}|g|\]
+\end{theorem}
+
+\begin{theorem}
+	If $f \in \mathcal{R}^*[a,b]$, the following assertions are equivalent:
+	\begin{enumerate}
+		\item $f \in \mathcal{L}[a,b]$.
+		\item There exists $\omega \in \mathcal{L}[a,b]$ such that $f(x)\leq\omega(x)$ for all $x \in [a,b]$.
+		\item There exists $\alpha \in \mathcal{L}[a,b]$ such that $\alpha(x) \leq f(x)$ for all $x \in [a,b]$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	If $f,g \in \mathcal{L}[a,b]$, then the functions $\max \{f,g\}$ and $\min \{f,g\}$ also belong to $\mathcal{L}[a,b]$.
+\end{theorem}
+
+\begin{theorem}
+	Suppose that $f,g,\alpha,$ and $\omega$ belong to $\mathcal{R}^*[a,b]$. If
+	\[f\leq\omega,\ g\leq\omega\ \ \text{  or if  }\ \ \alpha\leq f,\ \alpha \leq g,\]
+	then $\max \{f,g\}$ and $\min\{f,g\}$ also belong to $\mathcal{R}^*[a,b]$.
+\end{theorem}
+
+\begin{definition}
+	If $f \in \mathcal{L}[a,b]$, we define the \textbf{seminorm} of $f$ to be
+	\[||f||:=\displaystyle\int_{a}^{b}|f|\]
+	If $f,g \in \mathcal{L}[a,b]$, we define the \textbf{distance between $f$ and $g$} to be
+	\[\text{dist}(f,g):=||f-g||=\displaystyle\int_{a}^{b}|f-g|\]
+\end{definition}
+
+\begin{theorem}
+	The seminorm function satisfies:
+	\begin{enumerate}
+		\item $||f||\geq 0$ for all $f \in \mathcal{L}[a,b]$.
+		\item If $f(x)=0$ for $x \in [a,b]$, then $||f||=0$.
+		\item If $f \in \mathcal{L}[a,b]$ and $c \in \R$, then $||cf||=|c|\cdot||f||$.
+		\item If $f,g \in \mathcal{L}[a,b]$, then $||f+g||\leq||f||+||g||$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	The distance function satisfies:
+	\begin{enumerate}
+		\item $\dist(f,g)\geq 0$ for all $f,g \in \mathcal{L}[a,b]$.
+		\item If $f(x)=g(x)$ for $x \in [a,b]$, then $\dist(f,g)=0$.
+		\item $\dist(f,g)=\dist(g,f)$ for all $f,g \in \mathcal{L}[a,b]$.
+		\item $\dist(f,h)\leq\dist(f,g)+\dist(g,h)$ for all $f,g,h \in \mathcal{L}[a,b]$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}[\textbf{Completeness Theorem}]
+	A sequence $(f_n)$ of functions in $\mathcal{L}[a,b]$ converges to a function $f \in \mathcal{L}[a,b]$ if and only if it has the property that for every $\varepsilon>0$ there exists $H(\varepsilon)$ such that if $m,n\geq H(\varepsilon)$, then
+	\[||f_m-f_n||=\dist(f_m,f_n)<\varepsilon\]
+\end{theorem}

chapter-10/infinite-intervals.tex

@@ -0,0 +1,42 @@
+\section{Infinite Intervals}
+
+\begin{definition}
+	\begin{enumerate}
+		\item[]
+		\item A function $f:[a,\infty) \to \R$ is said to be  \textbf{generalized Riemann integrable} if there exists $A \in \R$ such that for every $\varepsilon>0$ there exists a gauge $\delta_\varepsilon$ on $[a,\infty]$ such that if $\dot{\mathcal{P}}$ is any $\delta_\varepsilon$-fine tagged subpartition of $[a,\infty)$, then $|S(f;\dot{\mathcal{P}})-A|\leq \varepsilon$. In this case, we write $f \in \mathcal{R}^*[a,\infty)$ and
+		      \[\displaystyle\int_{a}^{b}f:=A\]
+		\item A function $f:[a,\infty) \to \R$ is said to be \textbf{Lebesgue integrable} if both $f$ and $|f|$ belong to $\mathcal{R}^*[a,\infty)$. In this case we write $f \in \mathcal{L}[a,\infty)$.
+	\end{enumerate}
+\end{definition}
+
+\begin{theorem}[\textbf{Hake's Theorem}]
+	If $f:[a,\infty) \to \R$, then $f \in \mathcal{R}^*[a,\infty)$ if and only if for every $\gamma \in (a,\infty)$ the restriction of $f$ to $[a,\gamma]$ belongs to $\mathcal{R}^*[a,\gamma]$ and
+	\[\lim\limits_{\gamma\to\infty}\displaystyle\int_{a}^{\gamma}f=A\in\R\]
+	In this case $\displaystyle\int_{a}^{\infty}f=A$.
+\end{theorem}
+
+\begin{theorem}[\textbf{Fundamental Theorem}]
+	Suppose that $E$ is a countable subset of $[a,\infty)$ and that $f,F:[a,\infty)\to\R$ are such that:
+	\begin{enumerate}
+		\item $F$ is continuous on $[a,\infty)$ and $\lim\limits_{x \to \infty}F(x)$ exists.
+		\item $F'(x)=f(x)$ for all $x \in (a,\infty),\ x \notin E$.
+		      \\Then $f$ belongs to $\mathcal{R}^*[a,\infty)$ and
+		      \[\displaystyle\int_{a}^{\infty}f=\lim\limits_{x \to \infty}F(x)-F(a).\]
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}[\textbf{Hake's Theorem}]
+	If $h:(-\infty,\infty)\to\R$, then $h \in \mathcal{R}^*(-\infty,\infty)$ if and only if for every $\beta < \gamma$ in $(-\infty, \infty)$, the restriction of $h$ to $[\beta,\gamma]$ is in $\mathcal{R}^*[\beta,\gamma]$ and
+	\[\lim\limits_{\substack{\beta \to -\infty \\ \gamma \to +\infty}}\displaystyle\int_{\beta}^{\gamma}h=C\in\R\]
+	In this case $\displaystyle\int_{-\infty}^{\infty}h=C$.
+\end{theorem}
+
+\begin{theorem}[\textbf{Fundamental Theorem}]
+	Suppose that $E$ is a countable subset of $(-\infty, \infty)$ and that $h,H:(-\infty,\infty)\to\R$ satisfy:
+	\begin{enumerate}
+		\item $H$ is continuous on $(-\infty, \infty)$ and the limits $\lim\limits_{x \to \pm \infty}H(x)$ exist.
+		\item $H'(x)=h(x)$ for all $x \in (-\infty, \infty),\ x \notin E$.
+	\end{enumerate}
+	Then $h$ belongs to $\mathcal{R}^*(-\infty,\infty)$ and
+	\[\displaystyle\int_{-\infty}^{\infty}h=\lim\limits_{x \to \infty} H(x)-\lim\limits_{\gamma \to -\infty}H(\gamma)\]
+\end{theorem}

chapter-11/chapter-11.tex

@@ -0,0 +1,5 @@
+\chapter{A Glimpse Into Topology}
+\subimport{./}{open-and-closed-sets-in-R.tex}
+\subimport{./}{compact-sets.tex}
+\subimport{./}{continuous-functions.tex}
+\subimport{./}{metric-spaces.tex}

chapter-11/compact-sets.tex

@@ -0,0 +1,23 @@
+\section{Compact Sets}
+
+\begin{definition}
+	Let $A$ be a subset of $\R$. An \textbf{open cover} of $A$ is a collection $\mathcal{G}=\{G_\alpha\}$ of open sets in $\R$ whose union contains $A$; that is,
+	\[A \subseteq \bigcup_\alpha G_\alpha\]
+	If $\mathcal{G}'$ is a subcollection of sets from $\mathcal{G}$ such that the union of the sets in $\mathcal{G}'$ also contains $A$, then $\mathcal{G}'$ is called a \textbf{subcover} of $\mathcal{G}$. If $\mathcal{G}'$ consists of finitely many sets, then we call $\mathcal{G}'$ a \textbf{finite subcover} of $\mathcal{G}$.
+\end{definition}
+
+\begin{definition}
+	A subset $K$ of $\R$ is said to be \textbf{compact} if \textit{every} open cover of $K$ has a finite subcover.
+\end{definition}
+
+\begin{theorem}
+	If $K$ is a compact subset of $\R$, then $K$ is closed and bounded.
+\end{theorem}
+
+\begin{theorem}[\textbf{Heine-Borel Theorem}]
+	A subset $K$ of $\R$ is compact if and only if it is closed and bounded.
+\end{theorem}
+
+\begin{theorem}
+	A subset $K$ of $\R$ is compact if and only if every sequence in $K$ has a subsequence that converges to a point in $K$.
+\end{theorem}

chapter-11/continuous-functions.tex

@@ -0,0 +1,35 @@
+\section{Continuous Functions}
+
+\begin{lemma}
+	A function $f:A \to \R$ is continuous at the point $c$ in $A$ if and only if for every neighborhood $U$ of $f(c)$, there exists a neighborhood $V$ of $c$ such that if $x \in V\cap A$, then $f(x) \in U$.
+\end{lemma}
+
+\begin{theorem}[\textbf{Global Continuity Theorem}]
+	Let $A \subseteq \R$ and let $f:A \to \R$ be a function with domain $A$. Then the following are equivalent:
+	\begin{enumerate}
+		\item $f$ is continuous at every point of $A$.
+		\item For every open set $G$ in $\R$, there exists an open set $H$ in $\R$ such that $H \cap A=f^{-1}(G)$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{corollary}
+	A function $f:\R\to\R$ is continuous if and only if $f^{-1}(G)$ is open in $\R$ whenever $G$ is open.
+\end{corollary}
+
+\begin{theorem}[\textbf{Preservation of Compactness}]
+	If $K$ is a compact subset of $\R$ and if $f:K \to \R$ is continuous on $K$, then $f(K)$ is compact.
+\end{theorem}
+
+\begin{theorem}
+	If $K$ is a compact subset of $\R$ and $f:K \to \R$ is injective and continuous, then $f^{-1}$ is continuous on $f(K)$.
+\end{theorem}
+
+\begin{theorem}
+	Let $f:A \subseteq \R \to \R$. Then the following are equivalent:
+	\begin{enumerate}
+		\item $f$ is continuous on $A$
+		\item \textit{Theorem 5.1.3:} $\forall$ converging sequences $x_n \to a \in A$, then $f(x_n) \to f(a)$
+		\item \textit{Theorem 11.3.2:} For each open set $U \subseteq \R$, $f^{-1}(U) \subseteq A$ is open relative to $A$. (i.e. inverse images of open sets are relatively open to $A$. $f^{-1}(U)=V \cap A$ for some open set $V \subseteq \R$)
+		\item For each closed set $F \subseteq \R$, then $f^{-1}(F) \subseteq A$ is closed relative to $A$. (i.e. inverse images of closed sets are relatively closed)
+	\end{enumerate}
+\end{theorem}

chapter-11/metric-spaces.tex

@@ -0,0 +1,52 @@
+\section{Metric Spaces}
+
+\begin{definition}
+	A \textbf{metric} on a set $S$ is a function $d:S \times S \to \R$ that satisfies the following properties:
+	\begin{enumerate}
+		\item $d(x,y)\geq 0\ \forall\ x,y\in S$ \textit{(positivity)};
+		\item $d(x,y)=0$ if and only if $x=y$ \textit{(definiteness)};
+		\item $d(x,y)=d(y,x)\ \forall\ x,y \in S$ \textit{(symmetry)};
+		\item $d(x,y)\leq d(x,z)+d(z,y)\ \forall\ x,y,z\in S$ \textit{(triangle inequality)}
+	\end{enumerate}
+	A \textbf{metric space} $(S,d)$ is a set $S$ together with a metric $d$ on $S$.
+\end{definition}
+
+\begin{definition}
+	Let $(S,d)$ be a metric space. Then for $\varepsilon>0$, the $\varepsilon$\textbf{-neighborhood} of a point $x_0$ in $S$ is the set
+	\[V_\varepsilon(x_0):=\{x \in S :d(x_0,x)<\varepsilon\}\]
+	A \textbf{neighborhood} of $x_0$ is any set $U$ that contains an $\varepsilon$-neighborhood of $x_0$ for some $\varepsilon>0$.
+\end{definition}
+
+\begin{definition}
+	Let $(x_n)$ be a sequence in the metric space $(S,d)$. The sequence $(x_n)$ is said to \textbf{converge} to $x$ in $S$ if for any $\varepsilon>0$, there exists $K \in \N$ such that $x_n \in V_\varepsilon(x)$ for all $n \geq K$.
+\end{definition}
+
+\begin{definition}
+	Let $(S,d)$ be a metric space. A sequence $(x_n)$ in $S$ is said to be a \textbf{Cauchy sequence} if for each $\varepsilon>0$, there exists $H \in \N$ such that $d(x_n,x_m)<\varepsilon$ for all $n,m \geq H$.
+\end{definition}
+
+\begin{definition}
+	A metric space $(S,d)$ is said to be \textbf{complete} if each Cauchy sequence in $S$ converges to a point of $S$.
+\end{definition}
+
+\begin{definition}
+	Let $(S,d)$ be a metric space. A subset $G$ of $S$ is said to be an \textbf{open} set in $S$ if for every point $x \in S$ there is a neighborhood $U$ of $x$ such that $U \subseteq G$. A subset $F$ of $S$ is said to be a \textbf{closed} set in $S$ if the complement $S \setminus F$ is an open set in $S$.
+\end{definition}
+
+\begin{definition}
+	Let $(S_1,d_1)$ and $(S_2,d_2)$ be metric spaces, and let $f:S_1 \to S_2$ be a function from $S_1$ to $S_2$. The function $f$ is said to be \textbf{continuous} at the point $c$ in $S_1$ if for every $\varepsilon$-neighborhood $V_\varepsilon(f(c))$ of $f(c)$ there exists a $\delta$-neighborhood $V_\delta(c)$ of $c$ such that if $x \in V_\delta(c)$, then $f(x)\in V_\varepsilon(f(c))$.
+\end{definition}
+
+\begin{theorem}[\textbf{Global Continuity Theorem}]
+	If $(S_1,d_1)$ and $(S_2,d_2)$ are metric spaces, then a function $f:S_1 \to S_2$ is continuous on $S_1$ if and only if $f^{-1}(G)$ is open in $S_1$ whenever $G$ is open in $S_2$.
+\end{theorem}
+
+\begin{theorem}[\textbf{Preservation of Compactness}]
+	If $(S,d)$ is a compact metric space and if the function $f:S \to \R$ is continuous, then $f(S)$ is compact in $\R$.
+\end{theorem}
+
+\begin{definition}
+	A \textbf{semimetric} on a set $S$ is a function $d: S\times S \to \R$ that satisfies all of the conditions of a metric, except that condition (2) is replaces by the weaker condition
+	\[d(x,y)=0\ \ \text{  if  }\ \ x=y\]
+	A \textbf{semimetric space} $(S,d)$ is a set $S$ together with a semimetric $d$ on $S$.
+\end{definition}

chapter-11/open-and-closed-sets-in-R.tex

@@ -0,0 +1,49 @@
+\section{Open and Closed Sets in $\R$}
+
+\begin{definition}
+	A \textbf{neighborhood} of a point $x \in \R$ is any set $V$ that contains an $\varepsilon$-neighborhood $V_\varepsilon(x):=(x-\varepsilon,x+\varepsilon)$ of $x$ for some $\varepsilon>0$.
+\end{definition}
+
+\begin{definition}
+	\begin{enumerate}
+		\item[]
+		\item A subset $G$ of $\R$ is \textbf{open} in $\R$ if for each $x \in G$ there exists a neighborhood $V$ of $x$ such that $V \subseteq G$.
+		\item A subset $F$ of $\R$ is \textbf{closed in $\R$} if the complement $\mathcal{C}(F):=\R\setminus F$ is open in $\R$.
+	\end{enumerate}
+\end{definition}
+
+\begin{theorem}[\textbf{Open Set Properties}]
+	\begin{enumerate}
+		\item[]
+		\item The union of an arbitrary collection of open subsets in $\R$ is open.
+		\item The intersection of any finite collection of open sets in $\R$ is open.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}[\textbf{Closed Set Properties}]
+	\begin{enumerate}
+		\item[]
+		\item The intersection of an arbitrary collection of closed sets in $\R$ is closed.
+		\item The union of any finite collection of closed sets in $\R$ is closed.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}[\textbf{Characterization of Closed Sets}]
+	Let $F \subset \R$; then the following assertions are equivalent:
+	\begin{enumerate}
+		\item $F$ is a closed subset of $\R$.
+		\item If $X=(x_n)$ is any convergent sequence of elements in $F$, then $\lim X$ belongs to $F$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	A subset of $\R$ is closed if and only if it contains all of its cluster points.
+\end{theorem}
+
+\begin{theorem}
+	A subset of $\R$ is open if and only if it is the union of countably many disjoint open intervals in $\R$.
+\end{theorem}
+
+\begin{definition}
+	The \textbf{Cantor Set} $\mathbb{F}$ is the intersection of the sets $F_n, n\in\N$, obtained by successive removal of open middle thirds, starting with $[0,1]$.
+\end{definition}

chapter-2/absolute-value-and-the-real-line.tex

@@ -0,0 +1,41 @@
+\section{Absolute Value and the Real Line}
+
+\begin{definition}
+	The \textbf{absolute value} of a real number $a$, denoted by $|a|$, is defined by
+	\[|a|:=\begin{cases}
+			a  & \text{if } a>0, \\
+			0  & \text{if } a=0, \\
+			-a & \text{if } a<0.
+		\end{cases}\]
+\end{definition}
+
+\begin{theorem}
+	\begin{enumerate}
+		\item[]
+		\item $|ab|=|a||b|$ for all $a,b \in \R$.
+		\item $|a|^2 = a^2$ for all $a \in \R$.
+		\item If $c \geq 0$, then $|a| \leq c$ if and only if $-c \leq a \leq c$.
+		\item $-|a|\leq a \leq |a|$ for all $a \in \R$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}[\textbf{Triangle Inequality}]
+	If $a,b \in \R$, then $|a+b| \leq |a| + |b|$.
+\end{theorem}
+
+\begin{corollary}
+	If $a,b \in \R$, then
+	\begin{enumerate}
+		\item $\left| |a|-|b| \right| \leq |a-b|$,
+		\item $|a-b| \leq |a| + |b|$.
+	\end{enumerate}
+\end{corollary}
+
+\begin{corollary}
+	If $a_1, a_2, \dots, a_n$ are any real numbers, then
+	\[|a_1 + a_2 + \dots + a_n| \leq |a_1| + |a_2| + \dots + |a_n|\]
+\end{corollary}
+
+\begin{definition}
+	Let $a \in \R$ and $\varepsilon > 0$. Then the $\varepsilon$-\textbf{neighborhood} of $a$ is the set $V_\varepsilon(a):=\{x \in \R : |x-a| < \varepsilon\}$.
+\end{definition}

chapter-2/applications-of-the-supremum-property.tex

@@ -0,0 +1,29 @@
+\section{Applications of the Supremum Property}
+
+\begin{theorem}[\textbf{Archimedian Property}]
+	If $x \in \R$, then there exists $n_x \in \N$ such that $x \leq n_x$.
+\end{theorem}
+
+\begin{corollary}
+	If $S:= \{1/n : n \in \N\}$, then $\inf S = 0$.
+\end{corollary}
+
+\begin{corollary}
+	If $t >0$, there exists $n_t \in \N$ such that $0 < 1/n_t < t$.
+\end{corollary}
+
+\begin{corollary}
+	If $y>0$, there exists $n_y \in \N$ such that $n_y -1 \leq y \leq n_y$.
+\end{corollary}
+
+\begin{theorem}
+	There exists a positive real number $x$ such that $x^2 = 2$.
+\end{theorem}
+
+\begin{theorem}[\textbf{The Density Theorem}]
+	If $x$ and $y$ are any real numbers with $x<y$, then there exists a rational number $r \in \Q$ such that $x < r < y$.
+\end{theorem}
+
+\begin{corollary}
+	If $x$ and $y$ are real numbers with $x < y$, then there exists an irrational number $z$ such that $x < z < y$.
+\end{corollary}

chapter-2/chapter-2.tex

@@ -0,0 +1,6 @@
+\chapter{The Real Numbers}
+\subimport{./}{the-algebraic-and-order-properties-of-R.tex}
+\subimport{./}{absolute-value-and-the-real-line.tex}
+\subimport{./}{the-completeness-property-of-R.tex}
+\subimport{./}{applications-of-the-supremum-property.tex}
+\subimport{./}{intervals.tex}

chapter-2/intervals.tex

@@ -0,0 +1,45 @@
+\section{Intervals}
+
+\begin{definition}
+	If $a,b \in \R$ satisfy $a<b$, then the \textbf{open interval} determined by $a$ and $b$ is the set
+	\[(a,b):= \{x \in \R : a <x < b\}\]
+
+	The points $a$ and $b$ are called the \textbf{endpoints} of the interval.
+\end{definition}
+
+\begin{definition}
+	If both endpoints $a$ and $b$ are adjoined to an open interval, then we obtain the \textbf{closed interval} determined by $a$ and $b$; namely, the set
+	\[[a,b]:=\{x \in \R : a \leq x \leq b\}\]
+\end{definition}
+
+\begin{definition}
+	The two \textbf{half-open} (or \textbf{half-closed}) intervals determined by $a$ and $b$ are $[a,b)$, which includes the endpoint $a$, and $(a,b]$, which includes the endpoint $b$.
+\end{definition}
+
+\begin{definition}
+	The \textbf{length} of an interval $(a,b)$ is defined by $b-a$.
+\end{definition}
+
+\begin{theorem}[\textbf{Characterization Theorem}]
+	If $S$ is a subset of $\R$ that contains at least two points and has the property
+	\[\text{if}\ \ \ \ \ x,y \in S\ \ \ \ \ \text{and}\ \ \ \ \ x < y,\ \ \ \ \ \text{then}\ \ \ \ \  [x,y] \subseteq S,\]
+	then $S$ is an interval.
+\end{theorem}
+
+\begin{theorem}[\textbf{Nested Intervals Property}]
+	If $I_n=[a_n,b_n],\ n \in \N$, is a nested sequence of closed bounded intervals, then there exists a number $\xi \in \R$ such that $\xi \in I_n$ for all $n \in \N$.
+\end{theorem}
+
+\begin{theorem}
+	If $I_n :=[a_n,b_n],\ n \in \N$, is a nested sequence of closed, bounded intervals such that the lengths $b_n-a_n$ of $I_n$ satisfy
+	\[\inf \{b_n - a_n : n \in \N\}=0,\]
+	then the number $\xi$ contained in $I_n$ for all $n \in \N$ is unique.
+\end{theorem}
+
+\begin{theorem}
+	The set $\R$ of real numbers is not countable.
+\end{theorem}
+
+\begin{theorem}
+	The unit interval $[0,1] := \{x \in \R : 0 \leq x \leq 1\}$ is not countable.
+\end{theorem}

chapter-2/the-algebraic-and-order-properties-of-R.tex

@@ -0,0 +1,113 @@
+\section{The Algebraic and Order Properties of $\R$}
+
+\begin{theorem}[\textbf{Algebraic Properties of $\R$}]
+	On the set $\R$ of real numbers there are two binary operations, denoted by $+$ and $\cdot$ and called \textbf{addition} and \textbf{multiplication}, respectively. These operations satisfy the following properties:
+	\begin{enumerate}
+		\item[(A1)] $a+b=b+a\ \forall\ a,b \in \R$. (\textit{commutative property of addition});
+
+		\item[(A2)] $(a+b)+c=a+(b+c)\ \forall\ a,b,c \in \R$ (\textit{associative property of addition});
+
+		\item[(A3)] There exists and element $0$ in $\R$ such that $0+a=a$ and $a+0=a$ for all $a \in \R$ (\textit{existence of a zero element});
+
+		\item[(A4)] for each $a \in \R$ there exists and element $-a \in \R$ such that $a + (-a)=0$ and $(-a) + a=0$ (\textit{existence of negative elements});
+
+		\item[(M1)] $a \cdot b=b \cdot a\ \forall\ a,b \in \R$ (\textit{commutative property of multiplication});
+
+		\item[(M2)] $(a \cdot b) \cdot c = a \cdot (b \cdot c)\ \forall\ a,b,c \in \R$ (\textit{associative property of multiplication});
+
+		\item[(M3)] There exists an element $1 \in \R$ \textit{distinct from} $0$ such that $1 \cdot a=a$ and $a \cdot 1 = a\ \forall\ a \in \R$ (\textit{existence of a unit element});
+
+		\item[(M4)] for each $a \neq 0 \in \R$, there exists an element $1/a \in \R$ such that $a \cdot (1/a) = 1$ and $(1/a) \cdot a = 1$ (\textit{existence of reciprocals});
+
+			\item[(D)]$a \cdot (b+c)=(a \cdot b) + (a \cdot c)$ and $(b+c)\cdot a = (b \cdot a) + (c \cdot a)\ \forall\ a,b,c \in \R$ (\textit{distributive property of multiplication over addition}).
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	\begin{enumerate}
+		\item[]
+		\item If $z$ and $a$ are elements in $\R$ with $z+a=a$, then $z=0$.
+
+		\item If $u$ and $b \neq 0$ are elements in $\R$ with $u \cdot b=b$, then $u=1$.
+
+		\item If $a \in \R$, then $a \cdot 0=0$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	\begin{enumerate}
+		\item[]
+		\item If $a \neq 0$ and $b \in \R$ are such that $a \cdot b = 1$, then $b = 1/a$.
+
+		\item If $a \cdot b = 0$, then either $a=0$ or $b=0$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	There does not exists a rational number $r$ such that $r^2=2$.
+\end{theorem}
+
+\begin{definition}[\textbf{The Order Properties of $\R$}]
+	There is a nonempty subset $\mathbb{P}$ of $\R$, called the set of \textbf{positive real numbers}, that satisfies the following properties:
+	\begin{enumerate}
+		\item If $a,b$ belong to $\mathbb{P}$, then $a+b$ belongs to $\mathbb{P}$.
+
+		\item If $a,b$ belong to $\mathbb{P}$, then $ab$ belongs to $\mathbb{P}$.
+
+		\item If $a$ belongs to $\R$, then exactly one of the following holds:
+		      \[a \in \mathbb{P},\ \ \ \ a=0,\ \ \ \ -a \in \mathbb{P}\]
+		      (This condition is usually called the \textbf{Trichotomy Property}.)
+	\end{enumerate}
+\end{definition}
+
+\begin{definition}
+	Let $a,b$ be elements of $\R$.
+	\begin{enumerate}
+		\item If $a-b \in \mathbb{P}$, then we write $a >b$ or $b < a$.
+		\item If $a-b \in \mathbb{P} \cup \{0\}$, then we write $a \geq b$ or $b \leq a$.
+	\end{enumerate}
+\end{definition}
+
+\begin{theorem}
+	Let $a,b,c$ be any elements of $\R$.
+	\begin{enumerate}
+		\item If $a>b$ and $b>c$, then $a>c$.
+		\item If $a>b$, then $a+c>b+c$.
+		\item If $a>b$ and $c>0$, then $ca>cb$.
+		      \\If $a>b$ and $c<0$, then $ca<cb$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	\begin{enumerate}
+		\item[]
+		\item If $a \in \R$ and $a \neq 0$, then $a^2>0$.
+		\item $1 >0$.
+		\item If $n \in \N$, then $n >0$
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	If $a \in \R$ is such that $0 \leq a < \varepsilon$ for every $\varepsilon>0$, then $a=0$.
+\end{theorem}
+
+\begin{theorem}
+	If $ab>0$, then either
+	\begin{enumerate}
+		\item $a>0$ and $b>0$, or
+		\item $a<0$ and $b<0$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{corollary}
+	If $ab <0$, then either
+	\begin{enumerate}
+		\item $a<0$ and $b>0$, or
+		\item $a>0$ and $b<0$.
+	\end{enumerate}
+\end{corollary}
+
+\begin{definition}[\textbf{Bernoulli's Inequality}]
+	If $x>-1$, then
+	\[(1+x)^n \geq 1+nx\ \forall\ n \in \N\]
+\end{definition}

chapter-2/the-completeness-property-of-R.tex

@@ -0,0 +1,45 @@
+\section{The Completeness Property of $\R$}
+
+\begin{definition}
+	Let $S$ be a nonempty subset of $\R$.
+	\begin{enumerate}
+		\item The set $S$ is said to be \textbf{bounded above} if there exists a number $u \in \R$ such that $s \leq u$ for all $s \in S$. Each such number $u$ is called an \textbf{upper bound} of $S$.
+
+		\item The set $S$ is said to be \textbf{bounded below} if there exists a number $w \in \R$ such that $w \leq s$ for all $s \in S$. Each such number $w$ is called a \textbf{lower bound} of $S$.
+
+		\item A set is said to be \textbf{bounded} if it is both bounded above and bounded below. A set is said to be \textbf{unbounded} if it is not bounded.
+	\end{enumerate}
+\end{definition}
+
+\begin{definition}
+	Let $S$ be a nonempty subset of $\R$.
+	\begin{enumerate}
+		\item If $S$ is bounded above, then a number $u$ is said to be a \textbf{supremum} (or a \textbf{least upper bound}) of $S$ if it satisfies the conditions:
+		      \begin{enumerate}
+			      \item $u$ is an upper bound of $S$, and
+			      \item if $v$ is any upper bound of $S$, then $u \leq v$.
+		      \end{enumerate}
+
+		\item If $S$ is bounded below, then a number $w$ is said to be an \textbf{infimum} (or a \textbf{greatest lower bound}) of $S$ if it satisfies the conditions:
+		      \begin{enumerate}
+			      \item $w$ is a lower bound of $S$, and
+			      \item if $t$ is any lower bound of $S$, then $t \leq w$.
+		      \end{enumerate}
+	\end{enumerate}
+\end{definition}
+
+\begin{lemma}
+	A number $u$ is the supremum of a nonempty subset $S$ of $\R$ if and only if $u$ satisfies the conditions:
+	\begin{enumerate}
+		\item $s \leq u$ for all $s \in S$,
+		\item if $v < u$, then there exists $s' \in S$ such that $v < s'$.
+	\end{enumerate}
+\end{lemma}
+
+\begin{lemma}
+	An upper bound $u$ of a nonempty set $S$ in $\R$ is the supremum of $S$ if and only if for every $\varepsilon > 0$ there exists an $s_\varepsilon \in S$ such that $u - \varepsilon < s_\varepsilon$.
+\end{lemma}
+
+\begin{theorem}[\textbf{The Completeness Property of $\R$}]
+	Every nonempty set of real numbers that has an upper bound also has a supremum in $\R$. (This property is also called the \textbf{Supremum Property of $\R$}).
+\end{theorem}

chapter-3/chapter-3.tex

@@ -0,0 +1,8 @@
+\chapter{Sequences and Series}
+\subimport{./}{sequences-and-their-limits.tex}
+\subimport{./}{limit-theorems.tex}
+\subimport{./}{monotone-sequences.tex}
+\subimport{./}{subsequences-and-the-bolzano-wierstrass-theorem.tex}
+\subimport{./}{the-cauchy-criterion.tex}
+\subimport{./}{properly-divergent-sequences.tex}
+\subimport{./}{introduction-to-infinite-series.tex}

chapter-3/introduction-to-infinite-series.tex

@@ -0,0 +1,79 @@
+\section{Introduction to Infinite Series}
+
+\begin{definition}
+	If $X:=(x_n)$ is a sequence in $\R$, then the \textbf{infinite series} (or simply the \textbf{series}) \textbf{generated by} $X$ is the sequence $S:= (s_k)$ defined by
+	\begin{align*}
+		s_1 & := x_1                                \\
+		s_2 & := s_1 + s_2   & (=x_1 + x_2)         \\
+		    & \dots                                 \\
+		s_k & := s_{k-1}+x_k & (=x_1+x_2+\dots+x_k) \\
+		    & \dots
+	\end{align*}
+\end{definition}
+
+\begin{theorem}[\textbf{The $n$th Term Test}]
+	If the series $\sum x_n$ converges, then $\lim (x_n) = 0$.
+\end{theorem}
+
+\begin{theorem}[\textbf{Cauchy Criterion for Series}]
+	The series $\sum x_n$ converges if and only if for every $\varepsilon >0$ there exists $M(\varepsilon) \in \N$ such that if $m>n\geq M(\varepsilon)$, then
+	\[|s_m-s_n|=|x_{n+1}+x_{n+2}+\dots+x_m|<\varepsilon\]
+\end{theorem}
+
+\begin{theorem}
+	Let $(x_n)$ be a sequence of nonnegative real numbers. Then the series $\sum x_n$ converges if and only if the sequence $S=(s_k)$ of partial sums is bounded. In this case,
+	\[\sum\limits_{n=1}^{\infty}x_n = \lim (x_k) = \sup \{s_k:k \in \N\}\]
+\end{theorem}
+
+\begin{theorem}[\textbf{Comparison Test}]
+	Let $X:=(x_n)$ and $Y:=(y_n)$ be real sequences and suppose that for some $K \in \N$ we have
+	\[0 \leq x_n \leq y_n\ \ \ \text{for}\ \ \ n \geq K\]
+	\begin{enumerate}
+		\item Then the convergence of $\sum y_n$ implies the convergence of $\sum x_n$.
+		\item The divergence of $\sum x_n$ implies the divergence of $\sum y_n$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}[\textbf{Limit Comparison Test}]
+	Suppose that $X:=(x_n)$ and $Y:=(y_n)$ are strictly positive sequences and suppose that the following limit exists in $\R$:
+	\[r:=\lim \left(\frac{x_n}{y_n}\right)\]
+	\begin{enumerate}
+		\item If $r \neq 0$, then $\sum x_n$ is convergent if and only if $\sum y_n$ is convergent.
+		\item If $r=0$ and if $\sum y_n$ is convergent, then $\sum x_n$ is convergent.
+		\item If $r=\infty$ and $\sum y_n$ diverges, then $\sum x_n$ diverges.
+	\end{enumerate}
+\end{theorem}
+
+\begin{definition}
+	Let $(a_n):n \mapsto a(n)$ be a decreasing sequence of strictly positive terms in $\R$ which converges with a limit of zero. That is, for every $n$ in the domain of $(a_n):a_n>0,\ a_{n+1} \leq a_n$, and $a_n \to 0$ as $n \to + \infty$. The series $\displaystyle\sum_{n=1}^{\infty} 2^n a(2^n)$ is called the \textbf{condensed} form of the series $\displaystyle\sum_{n=1}^{\infty} a_n$.
+\end{definition}
+
+\begin{theorem}[\textbf{Cauchy Condensation Test}]
+	Let $(a_n):n \mapsto a(n)$ be a decreasing sequence of strictly positive terms in $\R$ which converges with a limit of zero. That is, for every $n$ in the domain of $(a_n):a_n>0,\ a_n \geq a_{n+1}$, and $a_n \to 0$ as $n \to +\infty$. Then the series $\displaystyle\sum_{n=1}^{\infty} a_n$ converges if and only if the condensed series $\displaystyle\sum_{n=1}^{\infty} 2^na(2^n)$ converges.
+\end{theorem}
+
+\begin{theorem}[\textbf{Cauchy Ratio Test}]
+	Let $\displaystyle\sum_{n=1}^{\infty} a_n$ be a series and $a_n>0$ for all $n \in \N$, and suppose the following limit exists in $\R$:
+	\[L=\lim\limits_{n \to \infty} \frac{a_{n+1}}{a_n}\]
+	\begin{enumerate}
+		\item If $L<1$, then the series converges absolutely.
+		\item If $L>1$, then the series is divergent.
+		\item If $L=1$ or the limit does not exist, then the test is inconclusive.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}[\textbf{Kummer's Test}]
+	Let $\sum a_n$ be a positive term series.
+	\begin{enumerate}
+		\item If there exists a positive term sequence $b_n$, $\alpha > 0$, and $\N \in \N$ such that $\displaystyle\frac{a_n}{a_{n+1}} \cdot b_n - b_{n+1} \geq \alpha,\ \forall\ n \geq N$, then $\sum a_n$ converges.
+		\item If $\displaystyle\frac{a_n}{a_{n+1}}\cdot b_n - b_{n+1} \leq 0,\ \forall\ n \geq N$, and if $\sum \frac{1}{b_n}$ diverges, then $\sum a_n$ diverges.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}[\textbf{Gauss' Test}]
+	If $\sum a_n$ is a positive term series, and if there exists a bounded sequence $b_n$ such that $\forall\ n \geq N$, $\displaystyle\frac{a_n}{a_{n+1}} = 1 +\frac{L}{n} + \frac{b_n}{n^2}$, then
+	\begin{enumerate}
+		\item If $L > 1$, then $\sum a_n$ converges.
+		\item If $L \leq 1$, then $\sum a_n$ diverges.
+	\end{enumerate}
+\end{theorem}

chapter-3/limit-theorems.tex

@@ -0,0 +1,50 @@
+\section{Limit Theorems}
+
+\begin{definition}
+	A sequence $X=(x_n)$ of real numbers is said to be \textbf{bounded} if there exists a real number $M>0$ such that $|x_n|\leq M$ for all $n \in \N$.
+	\\\\Thus, the sequence $(x_n)$ is bounded if and only if the set $\{x_n : n \in \N\}$ of its values is a bounded subset of $\R$.
+\end{definition}
+
+\begin{theorem}
+	A convergent sequence of real numbers is bounded.
+\end{theorem}
+
+\begin{theorem}
+	Let $X=(x_n)$ and $Y=(y_n)$ be sequences of real numbers that converge to $x$ and $y$, respectively, and let $c \in \R$.
+	\begin{enumerate}
+		\item Then the sequences $X+Y, X-Y, X \cdot Y$, and $cX$ converge to $x+y, x-y, xy$, and $cx$, respectively.
+
+		\item If $X=(x_n)$ converges to $x$ and $Z=(z_n)$ is a sequence of nonzero real numbers that converges to $z$ and if $z \neq 0$, then the quotient sequence $X/Z$ converges to $x/z$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	If $X=(x_n)$ is a convergent sequence of real numbers and if $x_n \geq 0$ for all $n \in \N$, then $x = \lim (x_n) \geq 0$.
+\end{theorem}
+
+\begin{theorem}
+	If $X=(x_n)$ and $Y=(y_n)$ are convergent sequences of real numbers and if $x_n \leq y_n$ for all $n \in \N$, then $\lim (x_n) \leq \lim (y_n)$.
+\end{theorem}
+
+\begin{theorem}
+	If $X=(x_n)$ is a convergent sequence and if $a \leq x_n \leq b$ for all $n \in \N$, then $a \leq \lim (x_n) \leq b$.
+\end{theorem}
+
+\begin{theorem}[\textbf{Squeeze Theorem}]
+	Suppose that $X=(x_n), Y= (y_n)$, and $Z=(z_n)$ are sequences of real numbers such that
+	\[x_n \leq y_n \leq z_n\ \ \ \text{for all }\ \ \ n \in \N\]
+	and that $\lim (x_n) = \lim (z_n)$. Then $Y=(y_n)$ is convergent and
+	\[\lim (x_n) = \lim (y_n) = \lim (z_n).\]
+\end{theorem}
+
+\begin{theorem}
+	Let the sequence $X=(x_n)$ converge to $x$. Then the sequence $(|x_n|)$ of absolute values converges to $|x|$. That is, if $x=\lim (x_n)$, then $|x|=\lim (|x_n|)$.
+\end{theorem}
+
+\begin{theorem}
+	Let $X=(x_n)$ be a sequence of real numbers that converges to $x$ and suppose that $x_n \geq 0$. Then the sequence $(\sqrt{x_n})$ of positive square roots converges and $\lim (\sqrt{x_n})=\sqrt{x}$.
+\end{theorem}
+
+\begin{theorem}
+	Let $(x_n)$ be a sequence of positive real numbers such that $L := \lim (x_{n+1}/x_n)$ exists. If $L <1$, then $(x_n)$ converges and $\lim (x_n)=0$.
+\end{theorem}

chapter-3/monotone-sequences.tex

@@ -0,0 +1,22 @@
+\section{Monotone Sequences}
+
+\begin{definition}
+	Let $X=(x_n)$ be a sequence of real numbers. We say that $X$ is \textbf{increasing} if it satisfies the inequalities
+	\[x_1 \leq x_2 \leq \dots \leq x_n \leq x_{n+1} \leq \dots\]
+
+	We say that $X$ is \textbf{decreasing} if it satisfies the inequalities
+	\[x_1 \geq x_2 \geq \dots \geq x_n \geq x_{n+1} \geq \dots\]
+
+	We say that $X$ is \textbf{monotone} if it is either increasing or decreasing.
+\end{definition}
+
+\begin{theorem}[\textbf{Monotone Convergence Theorem}]
+	A monotone sequence of real numbers is convergent if and only if it is bounded. Further:
+	\begin{enumerate}
+		\item If $X=(x_n)$ is a bounded increasing sequence, then
+		      \[\lim (x_n) = \sup \{x_n : n \in \N\}\]
+
+		\item If $Y=(y_n)$ is a bounded decreasing sequence, then
+		      \[\lim (y_n) = \inf \{y_n : n \in \N\}\]
+	\end{enumerate}
+\end{theorem}

chapter-3/properly-divergent-sequences.tex

@@ -0,0 +1,34 @@
+\section{Properly Divergent Sequences}
+
+\begin{definition}
+	Let $(x_n)$ be a sequence of real numbers.
+	\begin{enumerate}
+		\item We say that $(x_n)$ \textbf{tends to} $\pm \infty$, and write $\lim (x_n) = +\infty$, if for every $\alpha \in \R$ there exists a natural number $K(\alpha)$ such that if $n \geq K(\alpha)$, then $x_n > \alpha$.
+
+		\item We say that $(x_n)$ \textbf{tends to} $-\infty$, and write $\lim (x_n) = -\infty$, if for every $\beta \in \R$ there exists a natural number $K(\beta)$ such that if $n \geq K(\beta)$, then $x_n <  \beta$.
+	\end{enumerate}
+	We say that $(x_n)$ is \textbf{properly divergent} in case we have either $\lim (x_n)=+\infty$, or $\lim (x_n)=-\infty$.
+\end{definition}
+
+\begin{theorem}
+	A monotone sequence of real numbers is properly divergent if and only if it is unbounded.
+	\begin{enumerate}
+		\item If $(x_n)$ is an unbounded increasing sequence, then $\lim (x_n)=+\infty$.
+		\item If $(x_n)$ is an unbounded decreasing sequence, then $\lim (x_n) = -\infty$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	Let $(x_n)$ and $(y_n)$ be two sequences of real numbers and suppose that
+	\[x_n \leq y_n\ \ \ \text{for all}\ \ \ n \in \N\]
+	\begin{enumerate}
+		\item If $\lim (x_n) = +\infty$, then $\lim (y_n)=+\infty$.
+		\item If $\lim (y_n) = -\infty$, then $\lim (x_n)=-\infty$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	Let $(x_n)$ and $(y_n)$ be two sequences of positive real numbers and suppose that for some $L \in \R, L>0$, we have
+	\[\lim (x_n/y_n)=L\]
+	Then $\lim (x_n)=+\infty$ if an only if $\lim (y_n)=+\infty$.
+\end{theorem}

chapter-3/sequences-and-their-limits.tex

@@ -0,0 +1,39 @@
+\section{Sequences and Their Limits}
+
+\begin{definition}
+	A \textbf{sequence of real numbers} (or a \textbf{sequence in $\R$}) is a function defined on the set $\N = \{1,2,\dots\}$ of natural numbers whose range is contained in the set $\R$ of real numbers.
+\end{definition}
+
+\begin{definition}
+	A sequence $X = (x_n)$ in $\R$ is said to \textbf{converge} to $x \in \R$, or $x$ is said to be a \textbf{limit} of $(x_n)$, if for every $\varepsilon >0$ there exists a natural number $K(\varepsilon)$ such that for all $n \geq K(\varepsilon)$, the terms $x_n$ satisfy $|x_n-x|<\varepsilon$.
+	\\If a sequence has a limit, we say that the sequence is \textbf{convergent}; if it has no limit, we say that the sequence is \textbf{divergent}.
+\end{definition}
+
+\begin{theorem}[\textbf{Uniqueness of Limits}]
+	A sequence in $\R$ can have at most one limit.
+\end{theorem}
+
+\begin{theorem}
+	Let $X = (x_n)$ be a sequence of real numbers, and let $x \in \R$. The following statements are equivalent:
+	\begin{enumerate}
+		\item $X$ converges $x$.
+		\item For every $\varepsilon > 0$, there exists a natural number $K$ such that for all $n \geq K$, the terms $x_n$ satisfy $|x_n -x| < \varepsilon$.
+		\item For every $\varepsilon >0$, there exists a natural number $K$ such that for all $n \geq K$, the terms $x_n$ satisfy $x-\varepsilon < x_n < x+\varepsilon$.
+		\item For every $\varepsilon$-neighborhood $V_\varepsilon(x)$ of $x$, there exists a natural number $K$ such that for all $n \geq K$, the terms $x_n$ belong to $V_\varepsilon(x)$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{definition}
+	If $X=(x_1, x_2, \dots, x_n, \dots)$ is a sequence of real numbers and if $m$ is a given natural number, then the $m$-\textbf{tail} of $X$ is the sequence
+	\[X_m := (x_{m+n}: n \in \N)=(x_{m+1}, x_{m+2}, \dots)\]
+\end{definition}
+
+\begin{theorem}
+	Let $X=(x_n:n \in \N)$ be a sequence of real numbers and let $m \in \N$. Then the $m$-tail $X_m=(x_{m+n}:n \in \N)$ of $X$ converges if and only if $X$ converges. In this case, $\lim X_m = \lim X$.
+\end{theorem}
+
+\begin{theorem}
+	Let $(x_n)$ be a sequence of real numbers and let $x \in \R$. If $(a_n)$ is a sequence of positive real numbers with $\lim (a_n)=0$ and if for some constant $C >0$ and some $m \in \N$ we have
+	\[|x_n - x| \leq Ca_n\ \ \ \ \text{ for all }\ \ \ \ n \geq m\]
+	then it follows that $\lim (x_n) = x$.
+\end{theorem}

chapter-3/subsequences-and-the-bolzano-wierstrass-theorem.tex

@@ -0,0 +1,71 @@
+\section{Subsequences and the Bolzano-Wierstrass Theorem}
+
+\begin{definition}
+	Let $X=(x_n)$ be a sequence of real numbers and let $n_1 < n_2 < \dots < n_k < \dots$ be a strictly increasing sequence of natural numbers. Then the sequence $X' = (x_{n_k})$ given by
+	\[(x_{n_1}, x_{n_2}, \dots, x_{n_k}, \dots)\]
+	is called a \textbf{subsequence} of $X$.
+\end{definition}
+
+\begin{theorem}
+	If a sequence $X=(x_n)$ of real numbers converges to a real number $x$, then any subsequence $X' = (x_{n_k})$ of $X$ also converges to $x$.
+\end{theorem}
+
+\begin{theorem}
+	Let $X=(x_n)$ be a sequence of real numbers. Then the following are equivalent:
+	\begin{enumerate}
+		\item The sequence $X=(x_n)$ does not converge to $x \in \R$.
+
+		\item There exists an $\varepsilon_0 > 0$ such that for any $k \in \N$, there exists $n_k \in \N$ such that $n_k \geq k$ and $|x_{n_k}-x| \geq \varepsilon_0$.
+
+		\item There exists an $\varepsilon_0>0$ and a subsequence $X'=(x_{n_k})$ of $X$ such that $|x_{n_k}-x| \geq \varepsilon_0$ for all $k \in \N$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}[\textbf{Divergence Criteria}]
+	If a sequence $X=(x_n)$ of real numbers has either of the following properties, then $X$ is divergent.
+	\begin{enumerate}
+		\item $X$ has two convergent subsequences $X'=(x_{n_k})$ and $X''=(x_{r_k})$ whose limits are not equal.
+
+		\item $X$ is unbounded.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}[\textbf{Monotone Subsequence Theorem}]
+	If $X=(x_n)$ is a sequence of real numbers, then there is a subsequence of $X$ that is monotone.
+\end{theorem}
+
+\begin{theorem}[\textbf{The Bolzano-Wierstrass Theorem}]
+	A bounded sequence of real numbers has a convergent subsequence.
+\end{theorem}
+
+\begin{theorem}
+	Let $X=(x_n)$ be a bounded sequence of real numbers and let $x \in \R$ have the property that every convergent subsequence of $X$ converges to $x$. Then the sequence $X$ converges to $x$.
+\end{theorem}
+
+\begin{definition}
+	Let $X=(x_n)$ be a bounded sequence of real numbers.
+	\begin{enumerate}
+		\item The \textbf{limit superior} of $(x_n)$ is the infimum of the set $V$ of $v \in \R$ such that $v < x_n$ for at most a finite number of $n \in \N$. It is denoted by
+		      \[\lim \sup (x_n)\ \ \ \text{or}\ \ \ \lim \sup X\ \ \ \text{or}\ \ \ \overline{\lim} (x_n)\]
+
+		\item The \textbf{limit inferior} of $(x_n)$ is the supremum of the set of $w \in \R$ such that $x_m < w$ for at most a finite number of $m \in \N$. It is denoted by
+		      \[\lim \inf (x_n)\ \ \ \text{or}\ \ \ \lim \inf X\ \ \ \text{or}\ \ \ \overline{\lim}(x_n)\]
+	\end{enumerate}
+\end{definition}
+
+\begin{theorem}
+	If $(x_n)$ is a bounded sequence of real numbers, then the following statements for a real number $x^*$ are equivalent.
+	\begin{enumerate}
+		\item $x^* = \lim \sup (x_n)$.
+
+		\item If $\varepsilon>0$, there are at most a finite number of $n \in \N$ such that $x^* + \varepsilon < x_n$, but an infinite number of $n \in \N$ such that $x^*-\varepsilon < x_n$.
+
+		\item If $u_m=\sup \{x_n : n \geq m \}$, then $x^*= \inf \{u_m : m \in \N\}= \lim(u_m)$.
+
+		\item If $S$ is the set of subsequential limits of $(x_n)$, then $x^*= \sup S$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	A bounded sequence $(x_n)$ is convergent if and only if $\lim \sup (x_n)=\lim \inf (x_n)$.
+\end{theorem}

chapter-3/the-cauchy-criterion.tex

@@ -0,0 +1,36 @@
+\section{The Cauchy Criterion}
+
+\begin{theorem}
+	A sequence $X=(x_n)$ of real numbers is said to be a \textbf{Cauchy sequence} if for every $\varepsilon >0$ there exists a natural number $H(\varepsilon)$ such that for all natural numbers $n,m \geq H(\varepsilon)$, the terms $x_n, x_m$ satisfy $|x_n-x_m| < \varepsilon$.
+\end{theorem}
+
+\begin{lemma}
+	If $X=(x_n)$ is a convergent sequence of real numbers, then $X$ is a Cauchy sequence.
+\end{lemma}
+
+\begin{lemma}
+	A Cauchy sequence of real numbers is bounded.
+\end{lemma}
+
+\begin{theorem}[\textbf{Cauchy Convergence Criterion}]
+	A sequence of real numbers is convergent if and only if it is a Cauchy sequence.
+\end{theorem}
+
+\begin{definition}
+	We say that a sequence $X=(x_n)$ of real numbers is \textbf{contractive} if there exists a constant $C$, $0<C,1$, such that
+	\[|x_{n+2}-x_{n+1}| \leq C|x_{n+1}-x_n|\]
+	for all $n \in \N$. The number $C$ is called the \textbf{constant} of the contractive sequence.
+\end{definition}
+
+\begin{theorem}
+	Every contractive sequence is a Cauchy sequence, and therefore is convergent.
+\end{theorem}
+
+\begin{corollary}
+	If $X:=(x_n)$ is a contractive sequence with constant $C, 0<C<1$, and if $x^*:= \lim X$, then
+	\begin{enumerate}
+		\item $|x^*-x_n| \leq \frac{C^{n-1}}{1-C}|x_2-x_1|$,
+
+		\item $|x^*-x_n| \leq \frac{C}{1-C}|x_n-x_{n-1}|$.
+	\end{enumerate}
+\end{corollary}

chapter-4/chapter-4.tex

@@ -0,0 +1,4 @@
+\chapter{Limits}
+\subimport{./}{limits-of-functions.tex}
+\subimport{./}{limit-theorems.tex}
+\subimport{./}{some-extensions-of-the-limit-concept.tex}

chapter-4/limit-theorems.tex

@@ -0,0 +1,52 @@
+\section{Limit Theorems}
+
+\begin{definition}
+	Let $A \subseteq \R$, let $f:A \rightarrow \R$, and let $c \in \R$ be a cluster point of $A$. We say that $f$ is \textbf{bounded on a neighborhood of $c$} if there exists a $\delta$-neighborhood $V_\delta(c)$ of $c$ and a constant $M > 0$ such that we have $|f(x)| \leq M$ for all $x \in A \cap V_\delta (c)$.
+\end{definition}
+
+\begin{theorem}
+	If $A \subseteq \R$ and $f:A \rightarrow \R$ has a limit at $c \in \R$, then $f$ is bounded on some neighborhood of $c$.
+\end{theorem}
+
+\begin{theorem}
+	Let $A \subseteq \R$ and let $f$ and $g$ be functions defined on $A$ to $\R$. We  define the \textbf{sum} $f+g$, the \textbf{difference} $f-g$, and the \textbf{product} $fg$ on $A$ to $\R$ to be the functions given by
+	\[(f+g)(x):=f(x)+g(x),\]
+	\[(f-g)(x):=f(x)-g(x),\]
+	\[(fg)(x):=f(x)g(x)\]
+	for all $x \in A$. Further, if $b \in \R$, we define the \textbf{multiple} $bf$ to be the function given by
+	\[(bf)(x) := bf(x)\ \ \ \text{for all}\ \ \ x \in A\]
+	Finally, if $h(x)\neq 0$ for $x \in A$, we define the \textbf{quotient} $f/h$ to be the function given by
+	\[\left( \frac{f}{h}\right)(x) := \frac{f(x)}{h(x)}\ \ \ \ \text{for all}\ \ \ \ x \in A\]
+\end{theorem}
+
+\begin{theorem}
+	let $A \subseteq \R$, let $f$ and $g$ be functions on $A$ to $\R$, and let $c \in \R$ be a cluster point of $A$. Further, let $b \in \R$.
+	\begin{enumerate}
+		\item If $\lim\limits_{x\to c} f = L$ and $\lim\limits_{x\to c} g = M$, then
+		      \[\lim\limits_{x\to c} (f+g) = L+M,\]
+		      \[\lim\limits_{x\to c} (f-g)=L-M,\]
+		      \[\lim\limits_{x\to c} (fg) = LM,\]
+		      \[\lim\limits_{x\to c} (bf) = bL.\]
+
+		\item If $h: A \rightarrow \R$, if $h(x) \neq 0$ for all $x \in A$, and if $\lim\limits_{x\to c} h = H \neq 0$, then
+		      \[\lim\limits_{x\to c} \left( \frac{f}{h} \right)= \frac{L}{H}\]
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	Let $A \subseteq \R$, let $f: A \rightarrow \R$, and let $c \in \R$ be a cluster point of $A$. If
+	\[a \leq f(x) \leq b\ \ \ \ \text{for all}\ \ \ \ x \in A,\ x \neq c,\]
+	and if $\lim\limits_{x\to c} f$ exists, then $a \leq \lim\limits_{x\to c} f \leq b$.
+\end{theorem}
+
+\begin{theorem}[\textbf{Squeeze Theorem}]
+	Let $A \subseteq \R$, let $f,g,h:A \rightarrow \R$, and let $c \in \R$ be a cluster point of $A$. If
+	\[f(x) \leq g(x) \leq h(x)\ \ \ \ \text{for all}\ \ \ \ x \in A,\ x \neq c,\]
+	and if $\lim\limits_{x\to c} f = L = \lim\limits_{x\to c} h$, then $\lim\limits_{x\to c} g =L$.
+\end{theorem}
+
+\begin{theorem}
+	Let $A \subseteq \R$, let $f:A \rightarrow \R$ and let $c \in \R$ be a cluster point of $A$. If
+	\[\lim\limits_{x\to c} f > 0\ \ \ \left[\textit{respectively, } \lim\limits_{x\to c} f < 0\right],\]
+	then there exists a neighborhood $V_\delta (c)$ of $c$ such that $f(x) > 0$ [respectively, $f(x) < 0$] for all $x \in A \cap V_\delta (c),\ x \neq c$.
+\end{theorem}

chapter-4/limits-of-functions.tex

@@ -0,0 +1,50 @@
+\section{Limits of Functions}
+
+\begin{definition}
+	Let $A \subseteq \R$. A point $c \in \R$ is a \textbf{cluster point} of $A$ if for every $\delta >0$ there exists at least one point $x \in A,\ x \neq c$ such that $|x-c|<\delta$.
+	\\\\This definition is rephrased in the language of neighborhoods as follows: A point $c$ is a cluster point of the set $A$ if every $\delta$-neighborhood $V_\delta (c)=(c-\delta, c+\delta)$ of $c$ contains at least one point of $A$ distinct from $c$.
+\end{definition}
+
+\begin{theorem}
+	A number $c \in \R$ is a cluster point of a subset $A$ of $\R$ if and only if there exists a sequence $(a_n)$ in $A$ such that $\lim (a_n) = c$ and $a_n \neq c$ for all $n \in \N$.
+\end{theorem}
+
+\begin{definition}
+	Let $A \subseteq \R$, and let $c$ be a cluster point of $A$. For a function $f:A \rightarrow \R$, a real number $L$ is said to be a \textbf{limit of $f$ at $c$} if, given any $\varepsilon>0$, there exists a $\delta>0$ such that if $x \in A$ and $0 < |x-c|<\delta$, then $|f(x)-L|<\varepsilon$.
+\end{definition}
+
+\begin{theorem}
+	If $f:A \rightarrow \R$ and if $c$ is a cluster point of $A$, then $f$ can have only one limit at $c$.
+\end{theorem}
+
+\begin{theorem}
+	Let $f:A \rightarrow \R$ and let $c$ be a cluster point of $A$. Then the following statements are equivalent.
+	\begin{enumerate}
+		\item $\lim\limits_{x\to c}=L$.
+		\item Given any $\varepsilon$-neighborhood $V_\varepsilon (L)$ of $L$, there exists a $\delta$-neighborhood $V_\delta (c)$ of $c$ such that if $x \neq c$ is any point in $V_\delta (c) \cap A$, then $f(x)$ belongs to $V_\varepsilon (L)$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}[\textbf{Sequential Criterion}]
+	Let $f:A \rightarrow \R$ and let $c$ be a cluster point of $A$. Then the following are equivalent.
+	\begin{enumerate}
+		\item $\lim\limits_{x\to c} f=L$.
+		\item For every sequence $(x_n)$ in $A$ that converges to $c$ such that $x_n\neq c$ for all $n \in \N$, the sequence $(f(x_n))$ converges to $L$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}[\textbf{Divergence Criteria}]
+	Let $A \subseteq \R$, let $f:A \rightarrow \R$ and let $c \in \R$ be a cluster point of $A$.
+	\begin{enumerate}
+		\item If $L \in \R$, then $f$ does \textbf{not} have limit $L$ at $c$ if and only if there exists a sequence $(x_n)$ in $A$ with $x_n \neq c$ for all $n \in \N$ such that the sequence $(x_n)$ converges to $c$ but the sequence $(f(x_n))$ does \textbf{not} converge to $L$.
+
+		\item The function $f$ does \textbf{not} have a limit at $c$ if and only if there exists a sequence $(x_n)$ in $A$ with $x_n \neq c$ for all $n \in \N$ such that the sequence $(x_n)$ converges to $c$ but the sequence $(f(x_n))$ does \textbf{not} converge in $\R$.
+	\end{enumerate}
+\end{theorem}
+
+Let the \textbf{signum function} sgn be defined by
+\[\sign (x):=\begin{cases}
+		+1 & \text{for }  x>0,   \\
+		0  & \text{for }  x=0,   \\
+		-1 & \text{for }  x < 0.
+	\end{cases}\]

chapter-4/some-extensions-of-the-limit-concept.tex

@@ -0,0 +1,93 @@
+\section{Some Extensions of the Limit Concept}
+
+\begin{definition}
+	Let $A \in \R$ and let $f:A \rightarrow \R$.
+	\begin{enumerate}
+		\item If $c \in \R$ is a cluster point of the set $A \cap (c, \infty)= \{x \in A: x > c\}$, then we say that $L \in \R$ is a \textbf{right-hand limit of $f$ at $c$} and we write
+		      \[\lim\limits_{x\to c^+} f=L\ \ \ \ \ \text{or}\ \ \ \ \ \lim\limits_{x\to c^+} f(x)=L\]
+		      if given any $\varepsilon>0$ there exists a $\delta = \delta(\varepsilon)>0$ such that for all $x \in A$ with $0 < x-c < \delta$, then $|f(x)-L|<\varepsilon$.
+
+		\item If $c \in \R$ is a cluster point of the set $A \cap (-\infty, c)=\{x \in A: x <c\}$, then we say that $L \in \R$ is a \textbf{left-hand limit of $f$ at $c$} and we write
+		      \[\lim\limits_{x\to c^-} f = L\ \ \ \ \ \text{or}\ \ \ \ \ \lim\limits_{x\to c^-} f(x)=L\]
+		      if given any $\varepsilon > 0$ there exists a $\delta >0$ such that for all $x \in A$ with $0 < c-x < \delta$, then $|f(x)-L|<\varepsilon$.
+	\end{enumerate}
+\end{definition}
+
+\begin{theorem}
+	Let $A \subseteq \R$, let $f:A \rightarrow \R$, and let $c \in \R$ be a cluster point of $A \cap (c,\infty)$. Then the following statements are equivalent:
+	\begin{enumerate}
+		\item $\lim\limits_{x\to c^+} f = L$.
+
+		\item For every sequence $(x_n)$ that converges to $c$ such that $x_n \in A$ and $x_n > c$ for all $n \in \N$, the sequence $(f(x_n))$ converges to $L$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	Let $A \subseteq \R$, let $f:A \rightarrow \R$, and let $c \in \R$ be a cluster point of both of the sets $A \cap(c,\infty)$ and $A \cap (-\infty, c)$. Then $\lim\limits_{x\to c} f = L$ if and only if $\lim\limits_{x\to c^+} f = L = \lim\limits_{x\to c^-} f$.
+\end{theorem}
+
+\begin{theorem}
+	Let $A \subseteq \R$, let $f:A \rightarrow \R$, and let $c \in \R$ be a cluster point of $A$.
+	\begin{enumerate}
+		\item We say that $f$ \textbf{tends to $\infty$ as $x \rightarrow c$}, and write
+		      \[\lim\limits_{x\to c} f = \infty\]
+		      if for every $\alpha \in \R$ there exists $\delta = \delta(\alpha) > 0$ such that for all $x \in A$ with $0 < |x-c|<\delta$, then $f(x) > \alpha$.
+
+		\item We say that $f$ \textbf{tends to $-\infty$ as $x \rightarrow c$}, and write
+		      \[\lim\limits_{x\to c} f = - \infty\]
+		      if for every $\beta \in \R$ there exists $\delta = \delta (\beta)>0$ such that for all $x \in A$ with $0 < |x-c|<\delta$, then $f(x)<\beta$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	Let $A \subseteq \R$, let $f,g:A \rightarrow \R$, and let $c \in \R$ be a cluster point of $A$. Suppose that $f(x) \leq g(x)$ for all $x \in A,\ x \neq c$.
+	\begin{enumerate}
+		\item If $\lim\limits_{x\to c} f = \infty$, then $\lim\limits_{x\to c} g = \infty$.
+		\item If $\lim\limits_{x\to c} g = -\infty$, then $\lim\limits_{x\to c} f = -\infty$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{definition}
+	Let $A \subseteq \R$ and let $f:A \rightarrow \R$. If $c \in \R$ is a cluster point of the set $A \cap (c, \infty)= \{x \in A: x>c\}$, then we say that $f$ \textbf{tends to} $\infty$ [respectively, $-\infty$] as $x \rightarrow c^+$, and we write
+	\[\lim\limits_{x\to c^+} f = \infty\ \ \left[\text{respectively, } \lim\limits_{x\to c^+} f = -\infty\right]\]
+	if for every $\alpha \in \R$ there is $\delta = \delta(\alpha)>0$ such that for all $x \in A$ with $0 < x-c < \delta$, then $f(x)>\alpha$ [respectively, $f(x) < \alpha$]
+\end{definition}
+
+\begin{definition}
+	Let $A \subseteq \R$ and let $f:A \rightarrow \R$. Suppose that $(a,\infty) \subseteq A$ for some $a \in \R$. We say that $L \in \R$ is a \textbf{limit of $f$ as $x \rightarrow \infty$}, and write
+	\[\lim\limits_{x\to \infty} f = L\ \ \ \ \text{or}\ \ \ \ \lim\limits_{x\to \infty} f(x) = L,\]
+	if given any $\varepsilon > 0$ there exists $K=K(\varepsilon)>\alpha$ such that for any $x > K$, then $|f(x) - L| < \varepsilon$.
+\end{definition}
+
+\begin{theorem}
+	Let $A \subseteq \R$, let $f:A \rightarrow \R$, and suppose that $(a,\infty) \subseteq A$ for some $ a \in \R$. Then the following statements are equivalent:
+	\begin{enumerate}
+		\item $L=\lim\limits_{x\to \infty} f$.
+		\item For every sequence $(x_n)$ in $A \cap (a, \infty)$ such that $\lim (x_n) = \infty$, the sequence $(f(x_n))$ converges to $L$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{definition}
+	let $A \subseteq \R$ and let $f: A \rightarrow \R$. Suppose that $(a,\infty)\subseteq A$ for some $a \in A$. We say that $f$ \textbf{tends to} $\infty$ [respectively, $ - \infty$] \textbf{as} $x \rightarrow \infty$, and write
+	\[\lim\limits_{x\to \infty} f = \infty \ \ \ \ \ \left[\text{respectively, } \lim\limits_{x\to \infty} f = - \infty \right]\]
+	if given any $\alpha \in \R$ there exists $K = K(\alpha)>\alpha$ such that for any $x > K$, then $f(x)>\alpha$ [respectively, $f(x) < \alpha$].
+\end{definition}
+
+\begin{theorem}
+	Let $A \in \R$, let $f:A \rightarrow \R$, and suppose that $(a, \infty) \subseteq A$ for some $a \in \R$. Then the following statements are equivalent:
+	\begin{enumerate}
+		\item $\lim\limits_{x\to \infty} = \infty$ [respectively, $\lim\limits_{x\to \infty} f = - \infty$]
+
+		\item For every sequence $(x_n)$ in $(a, \infty)$ such that $\lim (x_n) = \infty$, then $\lim (f(x_n)) = \infty$ [respectively, $\lim (f(x_n)) = - \infty$].
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	Let $A \subseteq \R$, let $f,g:A \rightarrow \R$, and suppose that $(a,\infty) \subseteq A$ for some $a \in \R$. Suppose further that $g(x) > 0$ for all $x > a$ and that for some $L \in \R,\ L \neq 0$, we have
+	\[\lim\limits_{x\to \infty} \frac{f(x)}{g(x)} = L.\]
+	\begin{enumerate}
+		\item If $L > 0$, then $ \lim\limits_{x\to \infty} f = \infty$ if and only if $\lim\limits_{x\to \infty} g = \infty$.
+
+		\item If $L < 0$, then $\lim\limits_{x\to \infty} f = -\infty$ if and only if $\lim\limits_{x\to \infty} g = \infty$.
+	\end{enumerate}
+\end{theorem}

chapter-5/chapter-5.tex

@@ -0,0 +1,7 @@
+\chapter{Continuous Functions}
+\subimport{./}{continuous-functions.tex}
+\subimport{./}{combinations-of-continuous-functions.tex}
+\subimport{./}{continuous-functions-on-intervals.tex}
+\subimport{./}{uniform-continuity.tex}
+\subimport{./}{continuity-and-gauges.tex}
+\subimport{./}{monotone-and-inverse-functions.tex}

chapter-5/combinations-of-continuous-functions.tex

@@ -0,0 +1,45 @@
+\section{Combinations of Continuous Functions}
+
+\begin{theorem}
+	Let $A \subseteq \R$, let $f$ and $g$ be functions on $A$ to $\R$, and let $b \in \R$. Suppose that $c \in A$ and that $f$ and $g$ are continuous at $c$.
+	\begin{enumerate}
+		\item Then $f+g,\ f-g,\ fg$, and $bf$ are continuous at $c$.
+
+		\item If $h:A \rightarrow \R$ is continuous at $c \in A$ and if $h(x) \neq 0$ for all $x \in A$, then the quotient $f/h$ is continuous at $c$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	Let $A \subseteq \R$, let $f$ and $g$ be continuous on $A$ to $\R$, and let $b \in \R$.
+	\begin{enumerate}
+		\item The functions $f+g,\ f-g,\ fg$, and $bf$ are continuous on $A$.
+
+		\item If $h:A \rightarrow \R$ is continuous on $A$ and $h(x) \neq 0$ for $x \in A$, then the quotient $f/h$ is continuous on $A$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	Let $A \subseteq \R$, let $f:A \rightarrow \R$, and let $|f|$ be defined by $|f|(x) := |f(x)|$ for $x \in A$
+	\begin{enumerate}
+		\item If $f$ is continuous at at point $c \in A$, then $|f|$ is continuous at $c$.
+
+		\item If $f$ is continuous on $A$, then $|f|$ is continuous on $A$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	Let $A \subseteq \R$, let $f:A \rightarrow \R$, and let $f(x) \geq 0$ for all $x \in A$. We let $\sqrt{f}$ be defined for $x \in A$ by $(\sqrt{f})(x) := \sqrt{f(x)}$.
+	\begin{enumerate}
+		\item If $f$ is continuous at at point $c \in A$, then $\sqrt{f}$ is continuous at $c$.
+
+		\item If $f$ is continuous on $A$, then $\sqrt{f}$ is continuous on $A$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	Let $A,B \subseteq \R$ and let $f:A \rightarrow \R$ and $g:B \rightarrow \R$ be functions such that $f(A) \subseteq B$. If $f$ is continuous at a point $c \in A$ and g is continuous at $b= f(c) \in B$, then the composition $g \circ f:A \rightarrow \R$ is continuous $c$.
+\end{theorem}
+
+\begin{theorem}
+	Let $A,B \subseteq \R$, let $f:A \rightarrow \R$, be continuous on $A$, and let $g:B \rightarrow \R$ be continuous on $B$. If $f(A) \subseteq B$, then the composite function $g \circ f:A \rightarrow \R$ is continuous on $A$.
+\end{theorem}

chapter-5/continuity-and-gauges.tex

@@ -0,0 +1,23 @@
+\section{Continuity and Gauges}
+
+\begin{definition}
+	A \textbf{partition} of an interval $I := [a,b]$ is a collection $\mathbb{P} = \{I_1, \dots, I_n\}$ of non-overlapping closed intervals whose union is $[a,b]$. We ordinarily denote the intervals by $I_i:=[x_{i-1}, x_i]$, where
+	\[a=x_0 < \dots < x_{i-1} < x_i < \dots < x_n = b.\]
+	The points $x_i\ (i-0, \dots, n)$ are called the \textbf{partition points} of $\mathbb{P}$. If a point $t_i$ has been chosen from each interval $I_i$ for $i=1, \dots, n$, then the points $t_i$ are called the \textbf{tags} and the set of ordered pairs
+	\[\mathbb{P} = \{(I_1, t_1), \dots, (I_n,t_n)\}\]
+	is called a \textbf{tagged partition} of $I$. (The dot signifies that the partition is tagged.)
+\end{definition}
+
+\begin{definition}
+	A \textbf{gauge} on $I$ is a strictly positive function defined on $I$. If $\delta$ is a gauge on $I$, then a (tagged) partition $\mathbb{P}$ is said to be $\delta$-\textbf{fine} if
+	\[t_i \in I_i \subseteq [t_i - \delta(t_i), t_i + \delta(t_i)],\ \ \ \ \ \text{for}\ \ \ \ \ i=1, \dots, n.\]
+	We note that the notion of $\delta$-fineness requires that the partition be tagged, so we do not need to say ``tagged partition" in this case.
+\end{definition}
+
+\begin{lemma}
+	If a partition $\mathbb{P}$ of $I:=[a,b]$ is $\delta$-fine and $x \in I$, then there exists a tag $t_i$ in $\mathbb{P}$ such that $|x-t_i| \leq \delta (t_i)$.
+\end{lemma}
+
+\begin{theorem}
+	If $\delta$ is a gauge defined on the interval $[a,b]$, then there exists a $\delta$-fine partition of $[a,b]$.
+\end{theorem}

chapter-5/continuous-functions-on-intervals.tex

@@ -0,0 +1,43 @@
+\section{Continuous Functions on Intervals}
+
+\begin{definition}
+	A function $f:A \rightarrow \R$ is said to be \textbf{bounded on} $A$ if there exists a constant $M > 0$ such that $|f(x)| \leq M$ for all $x \in A$.
+\end{definition}
+
+\begin{theorem}[\textbf{Boundedness Theorem}]
+	Let $I:=[a,b]$ be a closed bounded interval and let $f: I \rightarrow \R$ be continuous on $I$. Then $f$ is bounded on $I$.
+\end{theorem}
+
+\begin{definition}
+	Let $A \subseteq \R$ and let $f:A \rightarrow \R$. We say that $f$ \textbf{has an absolute maximum} on $A$ if there is a point $x^* \in A$ such that
+	\[f(x^*) \geq f(x)\ \ \ \ \text{for all}\ \ \ \ x \in A.\]
+	We say that $f$ \textbf{has an absolute minimum} on $A$ if there is a point $x_* \in A$ such that
+	\[f(x_*) \leq f(x)\ \ \ \ \text{for all}\ \ \ \ x \in A.\]
+	We say that $x^*$ is an \textbf{absolute maximum point} for $f$ on $A$, and that $x_*$ is an \textbf{absolute minimum point} for $f$ on $A$, if they exist.
+\end{definition}
+
+\begin{theorem}[\textbf{Maximum-Minimum Theorem}]
+	Let $I := [a,b]$ be a closed bounded interval and let $f: I \rightarrow \R$ be continuous on $I$. Then $f$ has an absolute maximum and an absolute minimum on $I$.
+\end{theorem}
+
+\begin{theorem}[\textbf{Location of Roots Theorem}]
+	Let $I=[a,b]$ and let $f:I \rightarrow \R$ be continuous on $I$. If $f(a) < 0 < f(b)$, or if $f(a) > 0 > f(b)$, then there exists a number $c \in (a,b)$ such that $f(c)=0$.
+\end{theorem}
+
+\begin{theorem}[\textbf{Bolzano's Intermediate Value Theorem}]
+	Let $I$ be an interval and let $f:I \rightarrow \R$ be continuous on $I$. If $a,b \in I$ and if $k \in \R$ satisfies $f(a) < k<f(b)$, then there exists a point $c \in I$ between $a$ and $b$ such that $f(c) = k$.
+\end{theorem}
+
+\begin{corollary}
+	Let $I=[a,b]$ be a closed, bounded interval and let $f:I \rightarrow \R$ be on $I$. If $k \in \R$ is any number satisfying
+	\[\inf f(I) \leq k \leq \sup f(I),\]
+	then there exists a number $c \in I$ such that $f(c) = k$.
+\end{corollary}
+
+\begin{theorem}
+	Let $I$ be a closed bounded interval and let $f:I \rightarrow \R$ be continuous on $I$. Then the set $f(I):= \{f(x): x \in I\}$ is a closed bounded interval.
+\end{theorem}
+
+\begin{theorem}[\textbf{Preservation of Intervals Theorem}]
+	Let $I$ be an interval and let $f: I \rightarrow \R$ be continuous on $I$. Then the set $f(I)$ is an interval.
+\end{theorem}

chapter-5/continuous-functions.tex

@@ -0,0 +1,23 @@
+\section{Continuous Functions}
+
+\begin{definition}
+	Let $A \subseteq \R$, let $f:A \rightarrow \R$, and let $c \in A$. We say that $f$ is \textbf{continuous at} $c$ if, given any number $\varepsilon > 0$, there exists $\delta > 0$ such that if $x$ is any point of $A$ satisfying $|x-c|<\delta$, then $|f(x)-f(c)|<\varepsilon$.
+	\\If $f$ fails to be continuous at $c$, then we say that $f$ is \textbf{discontinuous at} $c$.
+\end{definition}
+
+\begin{theorem}
+	A function $f:A \rightarrow \R$ is continuous at a point $c \in A$ if and only if given any $\varepsilon$-neighborhood $V_\varepsilon (f(c))$ of $f(c)$ there exists a $\delta$-neighborhood $V_\varepsilon(c)$ of $c$ such that if $x$ is any point of $A \cap V_\delta(c)$, then $f(x)$ belongs to $V_\varepsilon (f(c))$, that is
+	\[f(A \cap V_\varepsilon (c)) \subseteq V_\varepsilon (f(c))\]
+\end{theorem}
+
+\begin{theorem}[\textbf{Sequential Criterion for Continuity}]
+	A function $f:A \rightarrow \R$ is continuous at the point $c \in A$ if and only if for every sequence $(x_n)$ in $A$ that converges to $c$, the sequence $(f(x_n))$ converges to $f(c)$.
+\end{theorem}
+
+\begin{theorem}[\textbf{Discontinuity Criterion}]
+	Let $A \subseteq \R$, let $f:A \rightarrow \R$, and let $c \in A$. Then $f$ is discontinuous at $c$ if and only if there exists a sequence $(x_n)$ in $A$ such that $(x_n)$ converges to $c$, but the sequence $(f(x_n))$ does not converge to $f(c)$.
+\end{theorem}
+
+\begin{definition}
+	Let $A \subseteq \R$ and let $f: A \rightarrow \R$. If $B$ is a subset of $A$, we say that $f$ is \textbf{continuous on the set} $B$ if $f$ is continuous at every point of $B$.
+\end{definition}

chapter-5/monotone-and-inverse-functions.tex

@@ -0,0 +1,53 @@
+\section{Monotone and Inverse Functions}
+
+\begin{definition}
+	Recall that if $A \subseteq \R$, then a function $f: A \to \R$ is said to be \textbf{increasing on} $A$ if whenever $x_1,x_2 \in A$ and $x_1 \leq x_2$, then $f(x_1) \leq f(x_2)$. The function $f$ is said to be \textbf{strictly increasing on} $A$ if whenever $x_1,x_2 \in A$ and $x_1<x_2$, then $f(x_1) < f(x_2)$. Similarly, $g:A \to \R$ is said to be \textbf{decreasing on} $A$ if whenever $x_1,x_2 \in A$ and $x_1 \leq x_2$ then $g(x_1) \geq g(x_2)$. The function $g$ is said to be \textbf{strictly decreasing on} $A$ if whenever $x_1, x_2 \in A$ and $x_1 < x_2$ then $g(x_1) > g(x_2)$.
+	\qquad If a function is either increasing or decreasing on $A$, we say that it is \textbf{monotone} on $A$. If $f$ is either strictly increasing or strictly decreasing on $A$, we say that $f$ is \textbf{strictly monotone} on $A$.
+\end{definition}
+
+\begin{theorem}
+	Let $I \subseteq \R$ be an interval and let $f:I \rightarrow \R$ be increasing on $I$. Suppose that $c \in I$ is not an endpoint of $I$. Then
+	\begin{enumerate}
+		\item $\lim\limits_{x\to c^-} f = \sup \{f(x): x \in I,\ x < c\}$,
+		\item $\lim\limits_{x\to c^+} f = \inf \{f(x): x \in I,\ x > c\}$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{corollary}
+	Let $I \subseteq \R$ be an interval and let $f:I \rightarrow \R$ be increasing on $I$. Suppose that $c \in I$ is not an endpoint of $I$. Then the following statements are equivalent.
+	\begin{enumerate}
+		\item $f$ is continuous at $c$.
+		\item $\lim\limits_{x\to c^-} f = f(c) = \lim\limits_{x\to c^+}$.
+		\item $\sup \{f(x):x \in I,\ x < c\} = f(c) = \inf \{f(x) : x \in I,\ x > c\}$.
+	\end{enumerate}
+\end{corollary}
+
+\begin{definition}
+	If $f:I \to \R$ is increasing on $I$ and if $c$ is not an endpoint of $I$, we define the \textbf{jump of $f$ at $c$} to be $j_f(c):=\lim\limits_{x \to c^+} f-\lim\limits_{x \to c^-} f$. It follows from \textit{Theorem 5.6.1} that
+	\[j_f(c)=\inf \{f(x)\ :\ x \in I,\ x > c\}- \sup \{f(x)\ :\ x \in I,\ x < c\}\]
+	for an increasing function. If the left endpoint $a$ of $I$ belongs to $I$, we define the \textbf{jump of $f$ at $a$} to be $j_f(a):=\lim\limits_{x \to a^+} f-f(a)$. If the right endpoint $b$ belongs to $I$, we define the \textbf{jump of $f$ at $b$} to be $j_f(b):=f(b)-\lim\limits_{x \to b^-} f$.
+\end{definition}
+
+\begin{theorem}
+	Let $I \subseteq \R$ be an interval and let $f:I \rightarrow \R$ be increasing on $I$. If $c \in I$, then $f$ is continuous at $c$ if and only if $j_f(c)=0$.
+\end{theorem}
+
+\begin{theorem}
+	Let $I \subseteq \R$ be an interval and let $f: I \rightarrow \R$ be monotone on $I$. Then the set of points $D \subseteq I$ at which $f$ is discontinuous is a countable set.
+\end{theorem}
+
+\begin{theorem}[\textbf{Continuous Inverse Theorem}]
+	Let $I \subseteq \R$ be an interval and let $f:I \rightarrow \R$ be strictly monotone and continuous on $I$. Then the function $g$ inverse to $f$ is strictly monotone and continuous on $J:=f(I)$.
+\end{theorem}
+
+\begin{definition}
+	\begin{enumerate}
+		\item[]
+		\item If $m,n \in \N$ and $x \geq 0$, we define $x^{m/n} := (x^{1/n})^m$.
+		\item If $m,n \in N$ and $x > 0$, we define $x^{-m/n} := (x^{1/n})^{-m}$.
+	\end{enumerate}
+\end{definition}
+
+\begin{theorem}
+	If $m \in \Z,\ n \in \N$, and $x > 0$, then $x^{m/n}=(x^m)^{1/n}$.
+\end{theorem}

chapter-5/uniform-continuity.tex

@@ -0,0 +1,66 @@
+\section{Uniform Continuity}
+
+\begin{definition}
+	Let $A \subseteq \R$ and let $f:A \rightarrow \R$. We say that $f$ is \textbf{uniformly continuous} on $A$ if for each $\varepsilon > 0$ there is a $\delta (\varepsilon) > 0$ such that if $x,u \in A$ are any numbers satisfying $|x-u|<\delta (\varepsilon)$, then $|f(x)-f(u)| < \varepsilon$.
+\end{definition}
+
+\begin{theorem}[\textbf{Nonuniform Continuity Criteria}]
+	Let $A \subseteq \R$ and let $f:A \rightarrow \R$. Then the following statements are equivalent:
+	\begin{enumerate}
+		\item $f$ is not uniformly continuous on $A$.
+
+		\item There exists an $\varepsilon_0 > 0$ such that for every $\delta > 0$ there are points $x_\delta, u_\delta$ in $A$ such that $|x_\delta - u_\delta|<\delta$ and $|f(x_\delta) - f(u_\delta)| \geq \varepsilon_0$.
+
+		\item There exists an $\varepsilon_0 > 0$ and two sequences $(x_n)$ and $(u_n)$ in $A$ such that $\lim (x_n - u_n)=0$ and $|f(x_n)-f(u_n)|\geq \varepsilon_0=1$ for all $n \in \N$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}[\textbf{Uniform Continuity Theorem}]
+	Let $I$ be a closed bounded interval and let $f:I \rightarrow \R$ be continuous on $I$. Then $f$ is uniformly continuous on $I$.
+\end{theorem}
+
+\begin{definition}
+	Let $A \subseteq \R$ and let $f:A \rightarrow \R$. If there exists a constant $K > 0$ such that
+	\[(4)\ \ \ \ \ \ \ \ |f(x)-f(u)| \leq K|x-u|\]
+	for all $x,u \in A$, then $f$ is said to be a \textbf{Lipschitz function} (or to satisfy a \textbf{Lipschitz condition}) on $A$.\\
+
+	The condition $(4)$ that a function $f: I \to \R$ on an interval $I$ is a Lipschitz function can be interpreted geometrically as follows. If we write the condition as
+	\[\abs{\frac{f(x)-f(u)}{x-u}}\leq K,\ x,u \in I,\ x \neq u,\]
+	then the quantity inside the absolute values is the slope of a line segment joining the points $(x,f(x))$ and $(u,f(u))$. Thus a function $f$ satisfies a Lipschitz condition if and only if the slopes of all line segments joining two points on the graph of $y=f(x)$ over $I$ are bounded by some number $K$.
+\end{definition}
+
+\begin{theorem}
+	If $f:A \rightarrow \R$ is a Lipschitz function, then $f$ is uniformly continuous on $A$.
+\end{theorem}
+
+\begin{theorem}
+	If $f:A \rightarrow \R$ is uniformly continuous on a subset $A$ of $\R$ and if $(x_n)$ is a Cauchy sequence in $A$, then $(f(x_n))$ is a Cauchy sequence in $\R$.
+\end{theorem}
+
+\begin{theorem}[\textbf{Continuous Extension Theorem}]
+	A function $f$ is uniformly continuous on the interval $(a,b)$ if and only if it can be defined at the endpoints $a$ and $b$ such that the extended function is continuous on $[a,b]$.
+\end{theorem}
+
+\begin{definition}
+	A function $s:[a,b] \rightarrow \R$ is called a \textbf{step function} if $[a,b]$ is the union of a finite number of nonoverlapping intervals $I_1, I_2, \dots, I_n$ such that $s$ is constant on each interval, that is, $s(x)=c_k$ for all $x \in I_k,\ k=1,2, \dots, n$.
+\end{definition}
+
+\begin{theorem}
+	Let $I$ be a closed bounded interval and let $f:I \rightarrow \R$ be continuous on $I$. If $\varepsilon > 0$, then there exists a step function $s_\varepsilon : I \rightarrow \R$ such that $|f(x) - s_\varepsilon(x)| < \varepsilon$ for all $x \in I$.
+\end{theorem}
+
+\begin{corollary}
+	Let $I:=[a,b]$ be a closed bounded interval and let $f: I \rightarrow \R$ be continuous on $I$. If $\varepsilon > 0$, there exists a natural number $m$ such that if we divide $I$ into $m$ disjoint intervals $I_k$ having length $h:= (b-a /m)$, then the step function $s_\varepsilon$ defined in equation (5) satisfies $|f(x) - s_\varepsilon (x)| < \varepsilon$ for all $x \in I$.
+\end{corollary}
+
+\begin{definition}
+	Let $I:= [a,b]$ be an interval. Then a function $g:I \rightarrow \R$ is said to be \textbf{piecewise linear} on $I$ if $I$ is the union of a finite number of disjoint intervals $I_1, \dots, I_m$, such that the restriction of $g$ to each interval $I_k$ is a linear function.
+\end{definition}
+
+\begin{theorem}
+	Let $I$ be a closed bounded interval and let $:I \rightarrow \R$ be continuous on $I$. If $\varepsilon > 0$, then there exists a continuous piecewise linear function $g_\varepsilon : I \rightarrow \R$ such that $|f(x) - g_\varepsilon (x)| < \varepsilon$ for all $ x \in I$.
+\end{theorem}
+
+\begin{theorem}[\textbf{Wierstrass Approximation Theorem}]
+	Let $I=[a,b]$ and let $f: I \rightarrow \R$ be a continuous function. If $\varepsilon > 0$ is given, then there exists a polynomial function $p_\varepsilon$ such that $|f(x) - p_\varepsilon (x)| < \varepsilon$ for all $ x \in I$.
+\end{theorem}

chapter-6/chapter-6.tex

@@ -0,0 +1,5 @@
+\chapter{Differentiation}
+\subimport{./}{the-derivative.tex}
+\subimport{./}{the-mean-value-theorem.tex}
+\subimport{./}{lhopitals-rules.tex}
+\subimport{./}{taylors-theorem.tex}

chapter-6/lhopitals-rules.tex

@@ -0,0 +1,32 @@
+\section{L'Hopital's Rules}
+
+\begin{theorem}
+	Let $f$ and $g$ be defined on $[a,b]$, let $f(a)=g(a)=0$, and let $g(x) \neq 0$ for $a < x < b$. If $f$ and $g$ are differentiable at $a$ and if $g'(a) \neq 0$, then the limit of $f/g$ at $a$ exists and is equal to $f'(a)/g'(a)$. Thus
+	\[\lim\limits_{x\to a^+} \frac{f(x)}{g(x)} = \frac{f'(a)}{g'(a)}\]
+\end{theorem}
+
+\begin{theorem}[\textbf{Cauchy Mean Value Theorem}]
+	Let $f$ and $g$ be continuous on $[a,b]$ and differentiable on $(a,b)$, and assume that $g'(x) \neq 0$ for all $x$ in $(a,b)$. Then there exists $c$ in $(a,b)$ such that
+	\[\frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f'(c)}{g'(c)}\]
+\end{theorem}
+
+\begin{theorem}[\textbf{L'Hopital's Rule, I}]
+	Let $-\infty \leq a < b \leq \infty$ and let $f,g$ be differentiable on $(a,b)$ such that $g'(x) \neq 0$ for all $x \in (a,b)$. Suppose that
+	\[\lim\limits_{x\to a^+} f(x) = 0 = \lim\limits_{x\to a^+} g(x)\]
+
+	\begin{enumerate}
+		\item If $\lim\limits_{x\to a^+} \frac{f'(x)}{g'(x)}=L \in \R$, then $\lim\limits_{x\to a^+} \frac{f(x)}{g(x)}=L$.
+
+		\item If $\lim\limits_{x\to a^+} \frac{f'(x)}{g'(x)}=L \in \{-\infty, \infty\}$, then $\lim\limits_{x\to a^+} \frac{f(x)}{g(x)}=L$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}[\textbf{L'Hopital's Rule, II}]
+	Let $-\infty \leq a < b \leq \infty$ and let $f,g$ be differentiable on $(a,b)$ such that $g'(x) \neq 0$ for all $x \in (a,b)$. Suppose that
+	\[\lim\limits_{x\to a^+} g(x) = \pm \infty\]
+	\begin{enumerate}
+		\item If $\lim\limits_{x\to a^+} \frac{f'(x)}{g'(x)}=L \in \R$, then $\lim\limits_{x\to a^+} \frac{f(x)}{g(x)}=L$.
+
+		\item If $\lim\limits_{x\to a^+} \frac{f'(x)}{g'(x)}=L \in \{-\infty, \infty\}$, then $\lim\limits_{x\to a^+} \frac{f(x)}{g(x)}=L$.
+	\end{enumerate}
+\end{theorem}

chapter-6/taylors-theorem.tex

@@ -0,0 +1,35 @@
+\section{Taylor's Theorem}
+
+\begin{theorem}[\textbf{Taylor's Theorem}]
+	Let $n \in \N$, let $I:=[a,b]$, and let $f:I \rightarrow \R$ be such that $f$ and its derivative $f', f'', \dots, f^{(n)}$ are continuous on $I$ and that $f^{(n+1)}$ exists on $(a,b)$. If $x_0\in I$, then for any $x$ in $I$ there exists a point $c$ between $x$ and $x_0$ such that
+	\[f(x) = f(x_0) + f'(x_0)(x-x_0) + \frac{f''(x_0)}{2!}(x-x_0)^2\]
+	\[+ \dots + \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n + \frac{f^{(n+1)}(c)}{(n+1)!}(x-x_0)^{n+1}\]
+\end{theorem}
+
+\begin{theorem}
+	Let $I$ be an interval, let $x_0$ be an interior point of $I$, and let $n \geq 2$. Suppose that the derivatives $f', f'', \dots, f^{(n)}$ exist and are continuous in a neighborhood of $x_0$ and that $f'(x_0) = \dots = f^{(n-1)(x_0)}$, but $f^{(n)}(x_0) \neq 0$.
+	\begin{enumerate}
+		\item If $n$ is even and $f^{(n)}(x_0) >0$, then $f$ has a relative minimum at $x_0$.
+
+		\item If $n$ is even and $f^{(n)}<0$, then $f$ has a relative maximum at $x_0$.
+
+		\item If $n$ is odd, then $f$ has neither a relative minimum nor relative maximum at $x_0$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{definition}
+	Let $I \subseteq \R$ be an interval. A function $f:I \rightarrow \R$ is said to be \textbf{convex} on $I$ if for any $t$ satisfying $0 \leq t \leq 1$ and any points $x_1, x_2$ in $I$, we have
+	\[f((1-t)x_1+tx_2) \leq (1-t)f(x_1)+tf(x_2).\]
+\end{definition}
+
+\begin{theorem}
+	Let $I$ be an open interval and let $f: I \rightarrow \R$ have a second derivative on $I$. Then $f$ is a convex function on $I$ if and only if $f''(x) \geq 0$ for all $x \in I$.
+\end{theorem}
+
+\begin{theorem}[\textbf{Newton's Method}]
+	Let $I:=[a,b]$ and let $f:I \rightarrow \R$ be twice differentiable on $I$. Suppose that $f(a)f(b) < 0$ and that there are constants $m,M$ such that $|f'(x)| \geq m > 0$ and $|f''(x)| \leq M$ for $x \in I$ and let $K:=M/2m$. Then there exists a subinterval $I^*$ containing a zero $r$ of $f$ such that for any $x_1 \in I^*$ the sequence $(x_n)$ defined by
+	\[|x_{n+1}-r|\leq x_n - \frac{f(x_n)}{f'(x_n)}\ \ \ \ \text{for all}\ \ \ \ n \in \N,\]
+
+	belongs to $I^*$ and $(x_n)$ converges to $r$. Moreover
+	\[|x_{n+1}-r| \leq K|x_n-r|^2\ \ \ \ \text{for all}\ \ \ \ n \in \N.\]
+\end{theorem}

chapter-6/the-derivative.tex

@@ -0,0 +1,63 @@
+\section{The Derivative}
+
+\begin{definition}
+	Let $I \subseteq \R$ be an interval, let $f:I \rightarrow \R$, and let $ c \in I$. We say that a real number $L$ is the \textbf{derivative of $f$ at $c$}  if given any $\varepsilon > 0$ there exists $\delta (\varepsilon) > 0$ such that if $x \in I$ satisfies $0 < |x-c|<\delta (\varepsilon)$, then
+	\[\abs{\frac{f(x)-f(c)}{x-c}-L}<\varepsilon.\]
+	In this case we say that $f$ is \textbf{differentiable} at $c$, and we write $f'(c)$ for $L$. In other words, the derivative of $f$ at $c$ is given by the limit
+	\[f'(c) = \lim\limits_{x\to c} \frac{f(x)-f(c)}{x-c}\]
+	provided this limit exists. (We allow the possibility that $c$ may be the endpoint of the interval.)
+\end{definition}
+
+\begin{theorem}
+	If $f:I \rightarrow \R$ has a derivative at $c \in I$, then $f$ is continuous at $c$.
+\end{theorem}
+
+\begin{theorem}
+	Let $I \subseteq \R$ be an interval, let $c \in I$ , and let $f:I \rightarrow \R$ and $g:I \rightarrow \R$ be functions that are differentiable at $c$. Then:
+	\begin{enumerate}
+		\item If $\alpha \in \R$, then the function $\alpha f$ is differentiable at $c$, and \[(\alpha f)'(c) = \alpha f'(c)\]
+
+		\item The function $f+g$ is differentiable at $c$, and
+		      \[(f+g)'(c) = f'(c)+g'(c)\]
+
+		\item (Product Rule) The function $fg$ is differentiable at $c$, and
+		      \[(fg)'(c) = f'(c)g(c) + f(c)g'(c).\]
+
+		\item (Quotient Rule) If $g(c) \neq 0$, then the function $f/g$ is differentiable at $c$, and
+		      \[\left( \frac{f}{g}\right)'(c) = \frac{f'(c)g(c)-f(c)g'(c)}{(g(c))^2}\]
+	\end{enumerate}
+\end{theorem}
+
+\begin{corollary}
+	If $f_1, f_2, \dots, f_n$ are functions on an interval $I$ to $\R$ that are differentiable at $c \in I$, then:
+	\begin{enumerate}
+		\item The function $f_1 + f_2 + \dots + f_n$ is differentiable at $c$ and
+		      \[(f_1 + f_2 + \dots + f_n)'(c) = f_1'(c) + f_2'(c) + \dots + f_n'(c)\]
+
+		\item The function $f_1f_2 \dots f_n$ is differentiable at $c$ and
+		      \[(f_1f_2 \dots f_n)'(c) = f_1'(c)f_2(c) \dots f_n(c)+f_1(c)f_2'(c) \dots f_n(c) + \dots + f_1(c)f_2(c) \dots f_n'(c).\]
+		      An important special case of the extended product rule occurs if the functions are equal, that is, $f_1 = f_2 = \dots = f_n = f$. Then the above becomes
+		      \[(f^n)'(c) = n(f(c))^{n-1}f'(c)\]
+	\end{enumerate}
+\end{corollary}
+
+\begin{theorem}[\textbf{Carathéodory's Theorem}]
+	Let $f$ be defined on an interval $I$ containing the point $c$. Then $f$ is differentiable at $c$ if and only if there exists a function $\varphi$ on $I$ that is continuous at $c$ and satisfies
+	\[f(x)-f(c)=\varphi (x)(x-c)\ \ \ \ \text{for}\ \ \ \ x \in I\]
+	In this case, we have $\varphi (c)=f'(c)$.
+\end{theorem}
+
+\begin{theorem}[\textbf{Chain Rule}]
+	Let $I, J$ be intervals in $\R$, let $g:I \rightarrow \R$ and $f:J \rightarrow \R$ be functions such that $f(J) \subseteq I$, and let $c \in J$. If $f$ is differentiable at $c$ and if $g$ is differentiable at $f(c)$, then the composite function $g \circ f$ is differentiable at $c$ and
+	\[(g \circ f)'(c) = g'(f(c)) \cdot f'(c).\]
+\end{theorem}
+
+\begin{theorem}
+	Let $I$ be an interval in $\R$ and let $f:I \rightarrow \R$ be strictly monotone and continuous on $I$. Let $J:=f(I)$ and let $g:J \rightarrow \R$ be the strictly monotone and continuous function inverse to $f$. If $f$ is differentiable at $c \in I$ and $f'(c) \neq0$, then $g$ is differentiable at $d:=f(c)$ and
+	\[g'(d)=\frac{1}{f'(c)}=\frac{1}{f'(g(d))}\]
+\end{theorem}
+
+\begin{theorem}
+	Let $I$ be an interval and let $f:I \rightarrow \R$ be strictly monotone on $I$. Let $J:= f(I)$ and let $g:J \rightarrow \R$ be the function inverse to $f$. If $f$ is differentiable on $I$ and $f'(x) \neq 0$ for $x \in I$, then $g$ is differentiable on $J$ and
+	\[g' = \frac{1}{f' \circ g}\]
+\end{theorem}

chapter-6/the-mean-value-theorem.tex

@@ -0,0 +1,56 @@
+\section{The Mean Value Theorem}
+
+\begin{theorem}[\textbf{Interior Extremum Theorem}]
+	Let $c$ be an interior point of of the interval $I$ at which $f:I \rightarrow \R$ has a relative extremum. If the derivative of $f$ at $c$ exists, then $f'(c)=0$.
+\end{theorem}
+
+\begin{corollary}
+	Let $f:I \rightarrow \R$ be continuous on an interval $I$ and suppose that $f$ has a relative extremum at an interior point $c$ of $I$. Then either the derivative of $f$ at $c$ does not exist, or it is equal to zero.
+\end{corollary}
+
+\begin{theorem}[\textbf{Rolle's Theorem}]
+	Suppose that $f$ is continuous on a closed interval $I:= [a,b]$, that the derivative $f'$ exists at every point of the open interval $(a,b)$, and that $f(a)=f(b)=0$. Then there exists at least one point $c$ in $(a,b)$ such that $f'(c)=0$.
+\end{theorem}
+
+\begin{theorem}[\textbf{Mean Value Theorem}]
+	Suppose that $f$ is continuous on a closed interval $I:=[a,b]$, and that $f$ has a derivative in the open interval $(a,b)$. Then there exists at least one point $c$ in $(a,b)$ such that
+	\[f(b)-f(a)=f'(c)(b-a)\]
+\end{theorem}
+
+\begin{theorem}
+	Suppose that $f$ is continuous on the closed interval $I:= [a,b]$, that $f$ is differentiable on the open interval $(a,b)$, and that $f'(x)=0$ for $x \in (a,b)$. Then $f$ is constant on $I$.
+\end{theorem}
+
+\begin{corollary}
+	Suppose that $f$ and $g$ are continuous on $I:=[a,b]$, that they are differentiable on $(a,b)$, and that $f'(x)=g'(x)$ for all $x \in (a,b)$. Then there exists a constant $C$ such that $f = g+C$ on $I$.
+\end{corollary}
+
+\begin{theorem}
+	Let $f:I \rightarrow \R$ be differentiable on the interval $I$. Then:
+	\begin{enumerate}
+		\item $f$ is increasing on $I$ if and only if $f'(x) \geq 0$ for all $x \in I$.
+		\item $f$ is decreasing on $I$ if and only if $f'(x) \leq 0$ for all $x \in I$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}[\textbf{First Derivative Test for Extrema}]
+	Let $f$ be continuous on the interval $I:=[a,b]$ and let $c$ be an interior point of $I$. Assume that $f$ is differentiable on $(a,c)$ and $(c,b)$. Then:
+	\begin{enumerate}
+		\item If there is a neighborhood $(c-\delta, c+\delta)\subseteq I$ such that $f'(x) \geq 0$ for $c-\delta< x < c$ and $f'(x) \leq 0$ for $c < x < c + \delta$, then $f$ has a relative maximum at $c$.
+
+		\item If there is a neighborhood $(c-\delta, c+\delta) \subseteq I$ such that $f'(x) \leq 0$ for $c-\delta < x < c$ and $f'(x) \geq 0$ for $c < x < c+\delta$, then $f$ has a relative maximum at $c$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{lemma}
+	Let $I \subseteq \R$ be an interval, let $f:I \rightarrow \R$, let $c \in I$, and assume that $f$ has a derivative at $c$. Then:
+	\begin{enumerate}
+		\item If $f'(c) >0$, then there is a number $\delta > 0$ such that $f(x) > f(c)$ for $x \in I$ such that $c < x < c+ \delta$.
+
+		\item If $f'(c)<0$, then there is a number $\delta >0$ such that $f(x) > f(c)$ for $x \in I$ such that $c-\delta < x < c$.
+	\end{enumerate}
+\end{lemma}
+
+\begin{theorem}[\textbf{Darboux's Theorem}]
+	If $f$ is differentiable on $I = [a,b]$ and if $k$ is a number between $f'(a)$ and $f'(b)$, then there is at least one point $c$ in $(a,b)$ such that $f'(c)=k$.
+\end{theorem}

chapter-7/approximate-integration.tex

@@ -0,0 +1,67 @@
+\section{Approximate Integration}
+
+\textbf{Equal Partitions}
+If $f:[a,b] \to \R$ is continuous, we know that its Riemann integral exists. To find an approximate value for this integral with the minimum amount of calculation, it is convenient to consider partitions $\mathcal{P}_n$ of $[a,b]$ into \textit{n equal} subintervals having length $h_n:=(b-a)/n$. Hence $\mathcal{P}_n$ is the partition:
+\[a<a+h_n<a+2h_n<\dots<a+nh_n=b\]
+If we pick our tag points to be the \textit{left endpoints} and the \textit{right endpoints} of the subintervals, we obtain the \textbf{$n$th left approximation} given by
+\[L_n(f):=h_n\sum\limits_{k=1}^{n-1}f(a+kh_n)\]
+and the \textbf{$n$th right approximation} given by
+\[R_n(f):=h_n\sum\limits_{k=1}^{n}f(a+kh_n)\]
+
+\begin{theorem}
+	If $f:[a,b] \to \R$ is monotone and if $T_n(f)$ is given by
+	\[T_n(f):=h_n\left(\frac{1}{2}f(a)+\sum\limits_{k=1}^{n-1}f(a+kh_n)+\frac{1}{2}f(b)\right)\]
+	then
+	\[\left|\displaystyle\int_{a}^{b}f-T_n(f)\right|\leq |f(b)-f(a)|\cdot \frac{(b-a)}{2n}\]
+\end{theorem}
+
+Note that the function $T_n(f)$ defined above is called the \textbf{$n$th Trapezoidal Approximation} of $f$
+
+\begin{theorem}
+	Let $f,f',$ and $f''$ be continuous on $[a,b]$ and let $T_n(f)$ be the $n$th Trapezoidal Approximation. Then there exists $c \in [a,b]$ such that
+	\[T_n(f)-\displaystyle\int_{a}^{b}f=\frac{(b-a)h^2_n}{12}\cdot f''(c)\]
+\end{theorem}
+
+\begin{corollary}
+	Let $f,f',$ and $f''$ be continuous, and let $|f''(x)|\leq B_2$ for all $x \in [a,b]$. Then
+	\[\left|T_n(f)-\displaystyle\int_{a}^{b}f\right|\leq \frac{(b-a)h_n^2}{12} \cdot B_2 = \frac{(b-a)^3}{12n^2}\cdot B_2\]
+\end{corollary}
+
+If $\mathcal{P}_n$ is the equally spaced partition given before, the \textbf{Midpoint Approximation} of $f$ is given by
+\[M_n(f):=h_n\left(f\left(a+\frac{1}{2}h_n\right)+f\left(a+\frac{3}{2}h_n\right)+\dots+f\left(a\left(n-\frac{1}{2}\right)h_n\right)\right)=h_n\sum\limits_{k=1}^{n}f\left(a+\left(k-\frac{1}{2}\right)h_n\right)\]
+
+\begin{theorem}
+	Let $f,f',$ and $f''$ be continuous on $[a,b]$ and let $M_n(f)$ be the $n$th Midpoint Approximation. Then there exists $\gamma \in [a,b]$ such that
+	\[\displaystyle\int_{a}^{b}f-M_n(f)=\frac{(b-a)h_n^2}{24}\cdot f''(\gamma)\]
+\end{theorem}
+
+\begin{corollary}
+	Let $f,f',$ and $f''$ be continuous, and let $|f''(x)| \leq B_2$ for all $x \in [a,b]$. Then
+	\[\left|M_n(f)-\displaystyle\int_{a}^{b}f\right|\leq \frac{(b-a)h_n^2}{24}\cdot B_2=\frac{(b-a)^3}{24n^2}\cdot B_2\]
+\end{corollary}
+
+The \textbf{$n$th Simpson Approximation} is defined by
+\[S_n(f):=\frac{1}{3}h_n(f(a)+4f(a+h_n)+2f(a+2h_n)+4f(a+3h_n)+2f(a+4h_n)+\dots+2f(b-2h_n)+4f(b-h_n)+f(b))\]
+
+\begin{theorem}
+	Let $f,f',f'',f^{(3)}$, and $f^{(4)}$ be continuous on $[a,b]$ and let $n \in \N$ be even. If $S_n(f)$ is the $n$th Simpson Approximation, then there exists $c \in [a,b]$ such that
+	\[S_n(f)-\displaystyle\int_{a}^{b}f=\frac{(b-a)h_n^4}{180}\cdot f^{(4)}(c)\]
+\end{theorem}
+
+\begin{corollary}
+	Let $f,f',f'',f^{(3)},$ and $f^{(4)}$ be continuous on $[a,b]$ and let $|f^{(4)}| \leq B_4$ for all $x \in [a,b]$. Then
+	\[\left|S_n(f)-\displaystyle\int_{a}^{b}f\right|\leq \frac{(b-a)h_n^4}{180}\cdot B_4 = \frac{(b-a)^5}{180n^4}\cdot B_4\]
+\end{corollary}
+
+\begin{remark}
+	The $n$th Midpoint Approximation $M_n(f)$ can be used to "step up" to the $(2n)$th Trapezoidal and Simpson Approximations by using the formulas
+	\[T_{2n}(f)=\frac{1}{2}M_n(f)+\frac{1}{2}T_n(f)\ \ \ \ \text{and}\ \ \ \ S_{2n}(f)=\frac{2}{3}M_n(f)+\frac{1}{3}T_n(f)\]
+	that are given in the exercises. Thus once the initial Trapezoidal Approximation $T_1=T_1(f)$ has been calculated, only the Midpoint Approximation $M_n=M_n(f)$ need be found. That is, we employ the following sequence of calculations:
+	\[T_1=\frac{1}{2}(b-a)(f(a)+f(b));\]
+	\begin{align*}
+		M_1=(b-a)f(\frac{1}{2}(a+b)),\ \ \ \  & T_2=\frac{1}{2}M_1+\frac{1}{2}T_1, & S_2=\frac{2}{3}M_1+\frac{1}{3}T_1; \\
+		M_2,\ \ \ \                           & T_4=\frac{1}{2}M_2+\frac{1}{2}T_2, & S_4=\frac{2}{3}M_2+\frac{1}{3}T_2; \\
+		M_4,\ \ \ \                           & T_8=\frac{1}{2}M_4+\frac{1}{2}T_4, & S_8=\frac{2}{3}M_4+\frac{1}{3}T_4; \\
+		\dots,\ \ \ \                         & \dots,                             & \dots
+	\end{align*}
+\end{remark}

chapter-7/chapter-7.tex

@@ -0,0 +1,6 @@
+\chapter{The Riemann Integral}
+\subimport{./}{riemann-integral.tex}
+\subimport{./}{riemann-integrable-functions.tex}
+\subimport{./}{the-fundamental-theorem.tex}
+\subimport{./}{the-darboux-integral.tex}
+\subimport{./}{approximate-integration.tex}

chapter-7/riemann-integrable-functions.tex

@@ -0,0 +1,54 @@
+\section{Riemann Integrable Functions}
+
+\begin{theorem}[\textbf{Cauchy Criterion}]
+	A function: $[a,b] \to \R$ belongs to $\mathcal{R}[a,b]$ if and only if for every $\varepsilon >0$ there exists $\eta_\varepsilon > 0$ such that if $\dot{\mathcal{P}}$ and $\dot{\mathcal{Q}}$ are any tagged partitions of $[a,b]$ with $||\dot{\mathcal{P}}||<\eta_\varepsilon$ and $||\dot{\mathcal{Q}}||<\eta_\varepsilon$, then
+	\[|S(f;\dot{\mathcal{P}})-S(f;\dot{\mathcal{Q}})|<\varepsilon\]
+\end{theorem}
+
+\begin{theorem}[\textbf{Squeeze Theorem}]
+	Let $f:[a,b] \to \R$. Then $f \in \mathcal{R}[a,b]$ if and only if for every $\varepsilon>0$ there exist functions $\alpha_\varepsilon$ and $\omega_\varepsilon$ in $\mathcal{R}[a,b]$ with
+	\[\alpha_\varepsilon(x) \leq f(x) \leq \omega_\varepsilon(x)\ \forall\ x \in [a,b]\]
+	and such that
+	\[\displaystyle\int_{a}^{b}(\omega_\varepsilon-\alpha_\varepsilon)<\varepsilon\]
+\end{theorem}
+
+\begin{lemma}
+	If $J$ is a subinterval of $[a,b]$ having endpoints $c < d$ and if $\varphi_J(x):=1$ for $x \in J$ and $\varphi_J(x):=0$ elsewhere in $[a,b]$, then $\varphi_J \in \mathcal{R}[a,b]$ and $\displaystyle\int_{a}^{b}\varphi_J=d-c$.
+\end{lemma}
+
+\begin{theorem}
+	If $\varphi:[a,b] \to \R$ is a step function, then $\varphi \in \mathcal{R}[a,b]$.
+\end{theorem}
+
+\begin{theorem}
+	If $f:[a,b] \to \R$ is continuous on $[a,b]$, then $f \in \mathcal{R}[a,b]$.
+\end{theorem}
+
+\begin{theorem}
+	If $f:[a,b]\to\R$ is monotone on $[a,b]$, then $f \in \mathcal{R}[a,b]$.
+\end{theorem}
+
+\begin{theorem}[\textbf{Additivity Theorem}]
+	Let $f:=[a,b] \to \R$ and let $c \in (a,b)$. Then $f \in \mathcal{R}[a,b]$ if and only if its restrictions to $[a,c]$ and $[c,b]$ are both Riemann integrable. In this case
+	\[\displaystyle\int_{a}^{b}f=\displaystyle\int_{a}^{c}f+\displaystyle\int_{c}^{b}f\]
+\end{theorem}
+
+\begin{corollary}
+	If $f \in \mathcal{R}[a,b]$, and if $[c,d]\subseteq [a,b]$, then the restriction of $f$ to $[c,d]$ is in $\mathcal{R}[c,d]$.
+\end{corollary}
+
+\begin{corollary}
+	If $f \in \mathcal{R}[a,b]$ and if $a=c_0<c_1<\dots<c_m=b$, then the restrictions of $f$ to each of the subintervals $[c_{i-1},c_i]$ are Riemann integrable and
+	\[\displaystyle\int_{a}^{b}f=\sum\limits_{i=1}^{m}\displaystyle\int_{c_{i-1}}^{c_i}f\]
+\end{corollary}
+
+\begin{definition}
+	If $f \in \mathcal{R}[a,b]$ and if $\alpha, \beta \in [a,b]$ with $\alpha < \beta$, we define
+	\[\displaystyle\int_{\beta}^{\alpha}f:=-\displaystyle\int_{\alpha}^{\beta}f\ \text{  and  }\ \displaystyle\int_{\alpha}^{\alpha}f:=0\]
+\end{definition}
+
+\begin{theorem}
+	If $f \in \mathcal{R}[a,b]$ and if $\alpha,\beta,\gamma$ are any numbers in $[a,b]$, then
+	\[\displaystyle\int_{\alpha}^{\beta}f=\displaystyle\int_{\alpha}^{\gamma}f+\displaystyle\int_{\gamma}^{\beta}f\]
+	in the sense that the existence of any two of these integrals implies the existence of the third integral and the equality.
+\end{theorem}

chapter-7/riemann-integral.tex

@@ -0,0 +1,53 @@
+\section{Riemann Integral}
+
+\begin{definition}
+	If $I:=[a,b]$ is a closed bounded interval in $\R$, then a \textbf{partition} of $I$ is a finite, ordered set $\mathcal{P}:=(x_0, x_1, \dots, x_{n-1}, x_n)$ of point in $I$ such that
+	\[a = x_0 < x_1 < \dots < x_{n-1} < x_n = b\]
+	Often we will denote the partition $\mathcal{P}$ by the notation $\mathcal{P}=\{[x_{i-1},x_i]\}_{i=1}^n$. We define the \textbf{norm} (or \textbf{mesh}) of $\mathcal{P}$ to be the number
+	\[||\mathcal{P}||:=\max\{x_1-x_0, x_2-x_1, \dots, x_n-x_{n-1}\}\]
+	Thus the norm {of a partition is merely the length of the largest subinterval into which the partition divides $[a,b]$. Clearly, many partitions have the same norm, so the partition is \textit{not}} a function of the norm.
+	\\If a point $t_i$ has been selected from each subinterval $I_i=[x_{i-1},x_i]$, for $i=1,2,\dots,n$, then the points are called \textbf{tags} of the subintervals of $I_i$. A set of ordered pairs
+	\[\dot{\mathcal{P}}:=\{([x_{i-1},x_i],t_i)\}_{i=1}^{n}\]
+	of subintervals and corresponding tags is called a \textbf{tagged partition} of $I$.
+	\\If $\dot{\mathcal{P}}$ is the tagged partition given above, we define the \textbf{Riemann sum} of a function $f:[a,b] \to \R$ corresponding to $\dot{\mathcal{P}}$ to be the number
+	\[S(f;\dot{\mathcal{P}}):=\sum\limits_{i=1}^{n} f(t_i)(x_i-x_{i-1})\]
+	We will also use this notation when $\dot{\mathcal{P}}$ denotes a \textit{subset} of a partition, and not the entire partition.
+\end{definition}
+
+\begin{definition}
+	A function $f:[a,b] \to \R$ is said to be \textbf{Riemann integrable} on $[a,b]$ if there exists a number $L \in \R$ such that for every $\varepsilon >0$ there exists $\delta_\varepsilon >0$ such that if $\dot{\mathcal{P}}$ is any tagged partition of $[a,b]$ with $||\dot{\mathcal{P}}||<\delta_\varepsilon$, then
+	\[|S(f;\dot{\mathcal{P}})-L|<\varepsilon\]
+	The set of all Riemann integrable functions on $[a,b]$ will be denoted by $\mathcal{R}[a,b]$.
+\end{definition}
+
+\begin{remark}
+	It is sometimes said that the integral $L$ is "the limit" of the Riemann sums $S(f:\dot{\mathcal{P}})$ as the norm $||\dot{\mathcal{P}}|| \to 0$. However, since $S(f;\dot{\mathcal{P}})$ is not a function of $||\dot{\mathcal{P}}||$, this \textbf{limit} is not of the type that we have studied before.
+	\\\\First we will show that if $f \in \mathcal{R}[a,b]$, then the number $L$ is uniquely determined. It will be called the \textbf{Riemann integral of $f$} over $[a,b]$. Instead of $L$, we will usually write
+	\[L=\int_{a}^{b}f\ \text{ or }\ \int_{a}^{b}f(x)dx\]
+\end{remark}
+
+\begin{theorem}
+	If $f \in \mathcal{R}[a,b]$, then the value of the integral is uniquely determined.
+\end{theorem}
+
+\begin{theorem}
+	If $g$ is Riemann integrable on $[a,b]$ and if $f(x)=g(x)$ except for a finite number of points in $[a,b]$, then $f$ is Riemann integrable and $\displaystyle\int_{a}^{b}f=\displaystyle\int_{b}^{a}g$.
+\end{theorem}
+
+\begin{theorem}
+	Suppose that $f$ and $g$ are in $\mathcal{R}[a,b]$. Then:
+	\begin{enumerate}
+		\item If $k \in \R$, the function $kf$ is in $\mathcal{R}[a,b]$ and
+		      \[\displaystyle\int_{a}^{b}kf=k\displaystyle\int_{a}^{b}f\]
+
+		\item The function $f+g$ is in $\mathcal{R}[a,b]$ and
+		      \[\displaystyle\int_{a}^{b}(f+g)=\displaystyle\int_{a}^{b}f+\displaystyle\int_{a}^{b}g\]
+
+		\item If $f(x) \leq g(x)$ for all $x \in [a,b]$, then
+		      \[\displaystyle\int_{a}^{b}f \leq \displaystyle\int_{a}^{b}g\]
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	If $f \in \mathcal{R}[a,b]$, then $f$ is bounded on $[a,b]$.
+\end{theorem}

chapter-7/the-darboux-integral.tex

@@ -0,0 +1,62 @@
+\section{The Darboux Integral}
+
+\begin{definition}[\textbf{Upper and Lower Sums}]
+	Let $f:I \to \R$ be a bounded function on $I=[a,b]$ and let $\mathcal{P}=(x_0,x_1,\dots,x_n)$ be a partition of $I$. for $k=1,2,\dots,n$ we let
+	\[m_k:=\inf \{f(x):x \in [x_{k-1},x_k]\},\ \ \ \ M_k:=\sup \{f(x):x \in [x_{k-1},x_k]\}\]
+	The \textbf{lower sum} of $f$ corresponding to the partition $\mathcal{P}$ is defined to be
+	\[L(f;\mathcal{P}):= \sum\limits_{k=1}^{n}m_k(x_k-x_{k-1})\]
+	and the \textbf{upper sum} of $f$ corresponding to $\mathcal{P}$ is defined to be
+	\[U(f;\mathcal{P}):=\sum\limits_{k=1}^{n}M_k(x_k-x_{k-1})\]
+\end{definition}
+
+\begin{lemma}
+	If $f:=I\to\R$ is bounded and $\mathcal{P}$ is any partition of $I$, then $L(f;\mathcal{P})\leq U(f;\mathcal{P})$.
+\end{lemma}
+
+\begin{definition}
+	If $\mathcal{P}:=(x_0,x_1,\dots,x_n)$ and $\mathcal{Q}:=(y_0,y_1,\dots,y_m)$ are partitions of $I$, we say that $\mathcal{Q}$ is a \textbf{refinement of } $\mathcal{P}$ if each partition point $x_k \in \mathcal{P}$ also belongs to $\mathcal{Q}$ (that is, if $\mathcal{P} \subseteq \mathcal{Q}$). A refinement $\mathcal{Q}$ of a partition $\mathcal{P}$ can be obtained by adjoining a finite number of points to $\mathcal{P}$.
+\end{definition}
+
+\begin{lemma}
+	If $f:I\to\R$ is bounded, if $\mathcal{P}$ is a partition of $I$, and if $Q$ is a refinement of $\mathcal{P}$, then
+	\[L(f;\mathcal{P})\leq L(f;\mathcal{Q})\ \text{ and }\ U(f;\mathcal{Q})\leq U(f;\mathcal{P})\]
+\end{lemma}
+
+\begin{lemma}
+	Let $f:I\to\R$ be bounded. If $\mathcal{P}_1,\mathcal{P}_2$ are any two partitions of $I$, then $L(f;\mathcal{P}_1)\leq U(f;\mathcal{P}_2)$.
+\end{lemma}
+
+\begin{definition}
+	We shall denote the collection of all partitions of the interval $I$ by $\mathscr{P}(I)$. Let $I:=[a,b]$ and let $f:I \to \R$ be a bounded function. The \textbf{lower integral of $f$ on $I$} is the number
+	\[L(f):=\sup\{L(f;\mathcal{P}):\mathcal{P} \in \mathscr{P}(I)\}\]
+	and the \textbf{upper integral of $f$ on $I$} is the number
+	\[U(f):=\inf\{U(f;\mathcal{P}):\mathcal{P} \in \mathscr{P}(I)\}\]
+\end{definition}
+
+\begin{theorem}
+	Let $I=[a,b]$ and let $f:I \to \R$ be a bounded function. Then the lower integral $L(f)$ and the upper integral $U(f)$ of $f$ on $I$ exist. Moreover,
+	\[L(f)\leq U(f)\]
+\end{theorem}
+
+\begin{definition}
+	Let $I=[a,b]$ and let $f:I \to \R$ be a bounded function. Then $f$ is said to be \textbf{Darboux integrable on $I$} if $L(f)=U(f)$. In this case the \textbf{Darboux Integral of $f$ over $I$} is defined to be the value $L(f)=U(f)$.
+\end{definition}
+
+\begin{theorem}[\textbf{Integrability Criterion}]
+	Let $I:=[a,b]$ and let $f:I \to \R$ be a bounded function on $I$. Then $f$ is Darboux integrable on $I$ if and only if for each $\varepsilon > 0$ there is a partition $\mathcal{P}_\varepsilon$ of $I$ such that
+	\[U(f;\mathcal{P}_\varepsilon)-L(f;\mathcal{P}_\varepsilon)<\varepsilon\]
+\end{theorem}
+
+\begin{corollary}
+	Let $I=[a,b]$ and let $f:I \to \R$ be a bounded function. If $\{P_n:n \in \N\}$ is a sequence of partitions on $I$ such that
+	\[\lim\limits_{n}(U(f;P_n)-L(f;P_n))=0,\]
+	then $f$ is integrable and $\lim\limits_{n}L(f;P_n)=\displaystyle\int_{a}^{b}f=\lim\limits_{n}U(f;P_n)$.
+\end{corollary}
+
+\begin{theorem}
+	If the function $f$ on the interval $I=[a,b]$ is either continuous or monotone on $I$, then $f$ is Darboux integrable on $I$.
+\end{theorem}
+
+\begin{theorem}[\textbf{Equivalence Theorem}]
+	A function $f$ on $I=[a,b]$ is Darboux integrable if and only if it is Riemann integrable.
+\end{theorem}

chapter-7/the-fundamental-theorem.tex

@@ -0,0 +1,82 @@
+\section{The Fundamental Theorem}
+
+\begin{theorem}[\textbf{Fundamental Theorem of Calculus (First Form)}]
+	Suppose there is a \textbf{finite} set $E$ in $[a,b]$ and functions $f,F:=[a,b] \to \R$ such that
+	\begin{enumerate}
+		\item $F$ is continuous on $[a,b]$,
+
+		\item $F'(x)=f(x)$ for all $x \in [a,b]\setminus E$,
+
+		\item $f$ belongs to $\mathcal{R}[a,b]$.
+	\end{enumerate}
+	Then we have
+	\[\displaystyle\int_{a}^{b}f=F(b)-F(a)\]
+\end{theorem}
+
+\begin{remark}
+	If the function $F$ is differentiable at every point of $[a,b]$, then (by \textit{Theorem 6.1.2}) hypothesis (a) is automatically satisfied. If $f$ is not defined for some point $c \in E$, we take $f(c):=0$. Even if $F$ is differentiable at every point of $[a,b]$, condition (c) is \textit{not automatically satisfied}, since there exists functions $F$ such that $F'$ is not Riemann integrable.
+\end{remark}
+
+\begin{definition}
+	If $f \in \mathcal{R}[a,b]$, then the function defined by
+	\[F(z):=\displaystyle\int_{a}^{z}f\ \text{for}\ z \in [a,b]\]
+	is called the \textbf{indefinite integral} of $f$ with \textbf{basepoint} $a$. (sometimes a point other than $a$ is used as a basepoint)
+\end{definition}
+
+\begin{theorem}
+	The indefinite integral $F$ defined by the above definition is continuous on $[a,b]$. In fact, if $|f(x)|\leq M$ for all $ x \in [a,b]$, then $|F(z)-F(w)|\leq M|z-w|$ for all $z,w \in [a,b]$.
+\end{theorem}
+
+\begin{theorem}[\textbf{Fundamental Theorem of Calculus (Second Form)}]
+	Let $f \in \mathcal{R}[a,b]$ and let $f$ be continuous at a point $c \in [a,b]$. Then the indefinite integral, defined by \textit{Definition 7.3.1}, is differentiable at $c$ and $F'(c)=f(c)$.
+\end{theorem}
+
+\begin{theorem}
+	If $f$ is continuous on $[a,b]$, then the indefinite integral $F$, defined by \textit{Definition 7.3.1}, is differentiable on $[a,b]$, and $F'(x)=f(x)$ for all $x \in [a,b]$.
+\end{theorem}
+
+\begin{theorem}[\textbf{Substitution Theorem}]
+	Let $J:=[\alpha, \beta]$ and let $\varphi:J\to\R$ have a continuous derivative on $J$. If $f:I\to\R$ is continuous on an interval $I$ containing $\varphi(J)$, then
+	\[\displaystyle\int_{\alpha}^{\beta}f(\varphi(t))\cdot\varphi'(t)dt=\displaystyle\int_{\varphi(\alpha)}^{\varphi(\beta)}f(x)dx\]
+\end{theorem}
+
+\begin{definition}
+
+	\begin{enumerate}
+		\item A set $Z \subset \R$ is said to be a \textbf{null set} if for every $\varepsilon>0$ there exists a countable collection $\{(a_k,b_k)\}_{k=1}^\infty$ of open intervals such that
+		      \[Z \subseteq \bigcup_{k=1}^{\infty}(a_k,b_k)\ \text{and}\ \sum\limits_{k=1}^{\infty}(b_k-a_k)\leq \varepsilon\]
+
+		\item If $Q(x)$ is a statement about the point $x \in I$, we say that $Q(x)$ holds \textbf{almost everywhere} on $I$ (or for \textbf{almost every} $x \in I$), if there exists a null set $Z \subset I$ such that $Q(x)$ holds for all $x \in I\setminus Z$. In this case, we may write
+		      \[Q(x)\ \text{for a.e. (almost everywhere)}\ x \in I\]
+	\end{enumerate}
+\end{definition}
+
+\begin{theorem}[\textbf{Lebesgue's Integrability Criterion}]
+	A bounded function $f:[a,b] \to \R$ is Riemann integrable if and only if it is continuous almost everywhere on $[a,b]$.
+\end{theorem}
+
+\begin{theorem}[\textbf{Composition Theorem}]
+	Let $f \in \mathcal{R}[a,b]$ with $f([a,b])\subseteq [c,d]$ and let $\varphi:[c,d] \to \R$ be continuous. Then the composition $\varphi \circ f$ belongs to $\mathcal{R}[a,b]$.
+\end{theorem}
+
+\begin{corollary}
+	Suppose that $f \in \mathcal{R}[a,b]$. Then its absolute value $|f|$ is in $\mathcal{R}[a,b]$, and
+	\[\left|\displaystyle\int_{a}^{b}f\right|\leq\displaystyle\int_{a}^{b}|f|\leq M(b-a),\]
+	where $|f(x)|\leq M$ for all $x \in [a,b]$.
+\end{corollary}
+
+\begin{theorem}[\textbf{The Product Theorem}]
+	If $f$ and $g$ belong to $\mathcal{R}[a,b]$, then the product $fg$ belongs to $\mathcal{R}[a,b]$.
+\end{theorem}
+
+\begin{theorem}[\textbf{Integration by Parts}]
+	Let $F,G$ be differentiable on $[a,b]$ and let $f:=F'$ and $g:=G'$ belong to $\mathcal{R}[a,b]$. Then
+	\[\left.\displaystyle\int_{a}^{b}fG=FG\ \right|_a^b-\displaystyle\int_{a}^{b}Fg\]
+\end{theorem}
+
+\begin{theorem}[\textbf{Taylor's Theorem with the Remainder}]
+	Suppose that $f',\dots,f^{(n)},f^{(n+1)}$ exist on $[a,b]$ and that $f^{(n+1)} \in \mathcal{R}[a,b]$. Then we have
+	\[f(b)=f(a)+\frac{f'(a)}{1!}(b-a)+\dots+\frac{f^{(n)}(a)}{n!}(b-a)^n+R_n,\]
+	where the remainder is given by
+	\[R_n=\frac{1}{n!}\displaystyle\int_{a}^{b}f^{(n+1)}(t)\cdot(b-t)^n dt\]
+\end{theorem}

chapter-8/chapter-8.tex

@@ -0,0 +1,5 @@
+\chapter{Sequences of Functions}
+\subimport{./}{pointwise-and-uniform-convergence.tex}
+\subimport{./}{interchange-of-limits.tex}
+\subimport{./}{the-exponential-and-logarithmic-functions.tex}
+\subimport{./}{the-trigonometric-functions.tex}

chapter-8/interchange-of-limits.tex

@@ -0,0 +1,30 @@
+\section{Interchange of Limits}
+
+\begin{theorem}
+	Let $(f_n)$ be a sequence of continuous functions on a set $A \subseteq \R$ and suppose that $(f_n)$ converges uniformly on $A$ to a function $f:A \to \R$. Then $f$ is continuous on $A$.
+\end{theorem}
+
+\begin{remark}
+	Although the uniform convergence of the sequence of continuous functions is sufficient to guarantee the continuity of the limit function, it is \textit{not} necessary.
+\end{remark}
+
+\begin{theorem}
+	Let $J \subseteq \R$ be a bounded interval and let $(f_n)$ be a sequence of functions on $J$ to $\R$. Suppose that there exists $x_0 \in J$ such that $(f_n(x_0))$ converges, and that the sequence $(f'_n)$ of derivatives exists on $J$ and converges uniformly on $J$ to a function $g$.
+	\\Then the sequence $(f_n)$ converges uniformly on $J$ to a function $f$ that has a derivative at every point of $J$ and $f'=g$.
+\end{theorem}
+
+\begin{theorem}
+	Let $(f_n)$ be a sequence of functions in $\mathcal{R}[a,b]$ and suppose that $(f_n)$ converges \textbf{uniformly} on $[a,b]$ to $f$. Then $f \in \mathcal{R}[a,b]$ and
+	\[\displaystyle\int_{a}^{b}f=\lim\limits_{n \to \infty}\displaystyle\int_{a}^{b}f_n\]
+	holds.
+\end{theorem}
+
+\begin{theorem}[\textbf{Bounded Convergence Theorem}]
+	Let $(f_n)$ be a sequence in $\mathcal{R}[a,b]$ that converges on $[a,b]$ to a function $f \in \mathcal{R}[a,b]$.. Suppose also that there exists $B >0$ such that $|f_n(x)|\leq B$ for all $x \in [a,b]$, $n \in \N$. Then
+	\[\displaystyle\int_{a}^{b}f=\lim\limits_{n \to \infty}\displaystyle\int_{a}^{b} f_n\]
+	holds.
+\end{theorem}
+
+\begin{theorem}[\textbf{Dini's Theorem}]
+	Suppose that $(f_n)$ is a monotone sequence of continuous functions on $I:=[a,b]$ that converges on $I$ to a continuous function $f$. Then the convergence of the sequence is uniform.
+\end{theorem}

chapter-8/pointwise-and-uniform-convergence.tex

@@ -0,0 +1,37 @@
+\section{Pointwise and Uniform Convergence}
+
+\begin{definition}
+	Let $(f_n)$ be a sequence of functions on $A \subseteq \R$ to $\R$, let $A_0\subseteq A$, and let $f: A_0 \to \R$. We say that the \textbf{sequence $(f_n)$ converges on $A_0$ to $f$} if, for each, $x \in A_0$, the sequence $(f_n(x))$ converges to $f(x)$ in $\R$. In this case we call $f$ the \textbf{limit on $A_0$ of the sequence $(f_n)$}. When such a function $f$ exists, we say that the sequence $(f_n)$ \textbf{is convergent on $A_0$}, or that $(f_n)$ \textbf{converges pointwise on $A_0$}.
+\end{definition}
+
+\begin{lemma}
+	A sequence $(f_n)$ of functions on $A \subseteq \R$ to $\R$ converges to a function $f:A_0 \to \R$ on $A_0$ if and only if for each $\varepsilon>0$ and each $x \in A_0$ there is a natural number $K(\varepsilon, x)$ such that if $n \geq K(\varepsilon, x)$, then
+	\[|f_n(x)-f(x)|<\varepsilon\]
+\end{lemma}
+
+\begin{definition}
+	A sequence $(f_n)$ of functions on $A \subseteq \R$ to $\R$ \textbf{converges uniformly on $A_0 \subseteq A$} to a function $f:A_0 \to \R$ if for each $\varepsilon >0$ there is a natural number $K(\varepsilon)$ (depending on $\varepsilon$ but \textbf{not} on $x \in A_0$) such that if $n \geq K(\varepsilon)$, then
+	\[|f_n(x)-f(x)|<\varepsilon\ \forall\ x \in A_0\]
+	In this case we say that the sequence $(f_n)$ is \textbf{uniformly convergent on $A_0$}. Sometimes we write
+	\[f_n \rightrightarrows f\ \text{on}\ A_0\ \text{or}\ f_n(x)\rightrightarrows f(x)\ \text{for}\ x \in A_0\]
+\end{definition}
+
+\begin{lemma}
+	A sequence $(f_n)$ of functions on $A \subseteq \R$ to $\R$ does not converge uniformly on $A_0 \subseteq A$ to a function $f:A_0 \to \R$ if and only if for some $\varepsilon_0 >0$ there is a subsequence $(f_{n_k})$ of $(f_n)$ and a sequence $(x_k)$ in $A_0$ such that
+	\[|f_{n_k}(x_k)-f(x_k)|\geq\varepsilon_0\ \forall\ k \in \N\]
+\end{lemma}
+
+\begin{definition}
+	If $A \subseteq \R$ and $\varphi : A \to \R$ is a function, we say that $\varphi$ is \textbf{bounded on $A$} if the set $\varphi(A)$ is a bounded subset of $\R$. If $\varphi$ is bounded we define the \textbf{uniform norm of $\varphi$ on $A$} by
+	\[||\varphi||_A:=\sup\{|\varphi(x)|:x \in A\}\]
+	Note that it follows that if $\varepsilon >0$, then
+	\[||\varphi||_A \leq \varepsilon \iff |\varphi(x)|\leq \varepsilon\ \forall\ x \in A\]
+\end{definition}
+
+\begin{lemma}
+	A sequence $(f_n)$ of bounded functions on $A\subseteq \R$ converges uniformly on $A$ to $f$ if and only if $||f_n - f||_A \to 0$.
+\end{lemma}
+
+\begin{theorem}[\textbf{Cauchy Criterion for Uniform Convergence}]
+	Let $(f_n)$ be a sequence of bounded functions on $A \subseteq \R$. Then this sequence converges uniformly on $A$ to a bounded function $f$ if and only if for each $\varepsilon>0$ there is a number $H(\varepsilon) \in \N$ such that for all $m,n\geq H(\varepsilon)$, then $||f_m-f_n||_A \leq \varepsilon$.
+\end{theorem}

chapter-8/the-exponential-and-logarithmic-functions.tex

@@ -0,0 +1,92 @@
+\section{The Exponential and Logarithmic Functions}
+
+\begin{theorem}
+	There exists a function $:\R \to \R$ such that:
+	\begin{enumerate}
+		\item $E'(x)=E(x)\ \forall\ x \in \R$.
+		\item $E(0)=1$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{corollary}
+	The function $E$ has a derivative of every order and $E^{(n)}(x)=E(x)$ for all $n \in \N$, $x \in \R$.
+\end{corollary}
+
+\begin{corollary}
+	If $x>0$, then $1+x < E(x)$.
+\end{corollary}
+
+\begin{theorem}
+	The function $E:\R\to\R$ that satisfies (1) and (2) of \textit{Theorem 8.3.1} is unique.
+\end{theorem}
+
+\begin{theorem}
+	The unique function $E:\R\to\R$, such that $E'(x)=E(x)$ for all $x \in \R$ and $E(0)=1$, is called the \textbf{exponential function}. The number $e=E(1)$ is called \textbf{Euler's number}. We will frequently write
+	\[\exp(x):=E(x)\ \text{or}\  e^x:=E(x)\ \text{for}\ x \in \R\]
+\end{theorem}
+
+\begin{theorem}
+	The exponential function satisfies the following properties:
+	\begin{enumerate}
+		\item $E(x) \neq 0$ for all $x \in \R$;
+		\item $E(x_+y)=E(x)E(y)$ for all $x,y,\in\R$.
+		\item $E(r) = e^r$ for all $r \in \Q$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	The exponential function $E$ is strictly increasing on $\R$ and has range equal to $\{y \in \R : y > 0\}$. Further, we have
+	\[\lim\limits_{x \to -\infty} E(x)=0\ \ \text{and}\ \ \lim\limits_{x \to \infty} = \infty\]
+\end{theorem}
+
+\begin{definition}
+	The function inverse to $E:\R \to \R$ is called the \textbf{logarithm} (or the \textbf{natural logarithm}). It will be denoted by $L$, or by $\ln$.
+\end{definition}
+
+\begin{theorem}
+	The logarithm is a strictly increasing function $L$ with domain $\{x \in \R : x > 0\}$ and range $\R$. The derivative of $L$ is given by
+	\begin{enumerate}
+		\item $L'(x)=1/x$ for $x >0$.The logarithm satisfies the functional equation
+		\item $L(xy)=L(x)+L(y)$ for $x>0,  y>0$. Moreover, we have
+		\item $L(1)=0$ and $L(e)=1$,
+		\item $L(x^r)=rL(x)$ for $x > 0$, $r \in \Q$,
+		\item $\lim\limits_{x \to 0^+} L(x)=-\infty$ and $\lim\limits_{x \to \infty}L(x) = \infty$
+	\end{enumerate}
+\end{theorem}
+
+\begin{definition}
+	If $\alpha \in \R$ and $x > 0$, the number $x^\alpha$ is defined to be
+	\[x^\alpha := e^{\alpha \ln x}=E(\alpha L(x))\]
+	The function $x \mapsto x^\alpha$ for $x > 0$ is called the \textbf{power function} with exponent $\alpha$.
+\end{definition}
+
+\begin{theorem}
+	If $\alpha \in \R$ and $x,y$ belong to $(0, \infty)$, then:
+	\begin{enumerate}
+		\item $1^\alpha = 1$
+		\item $x^\alpha >0$
+		\item $(xy)^\alpha = x^\alpha y^\alpha$
+		\item $(x/y)^\alpha = x^\alpha / y^\alpha$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	If $\alpha, \beta \in \R$ and $x \in (0,\infty)$, then:
+	\begin{enumerate}
+		\item $x^{\alpha + \beta}=x^\alpha x^\beta$
+		\item $(x^\alpha)^\beta = x^{\alpha \beta}=(x^\beta)^\alpha$
+		\item $x^{-\alpha} = 1/x^\alpha$
+		\item if $\alpha < \beta$, then $x^\alpha < x^\beta$ for $x > 1$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	Let $\alpha \in \R$. Then the function $x \mapsto x^\alpha$ on $(0,\infty)$ to $\R$ is continuous and differentiable and
+	\[Dx^\alpha = \alpha x^{\alpha - 1}\ \ \text{for}\ \ x \in (0, \infty)\]
+\end{theorem}
+
+\begin{definition}
+	Let $a>0,\ a \neq 1$. We define
+	\[\log_a(x) := \frac{\ln(x)}{\ln(a)}\ \ \text{for}\ \ x \in (0,\infty)\]
+	For $x \in (0,\infty)$, the number $\log_a(x)$ is called the \textbf{logarithm of $x$ to the base $a$}. The case $a=e$ yields the logarithm (or natural logarithm) function of \textit{Definition 8.3.1}. The case $a=10$ give sthe base 10 logarithm (or common logarithm) function $\log_{10}$ often used in computations. Properties of the functions $\log_a$ will be given in the exercises.
+\end{definition}

chapter-8/the-trigonometric-functions.tex

@@ -0,0 +1,69 @@
+\section{The Trigonometric Functions}
+
+\begin{theorem}
+	There exist functions $C:\R \to \R$ and $S:\R\to\R$ such that
+	\begin{enumerate}
+		\item $C''(x)=-C(x)$ and $S''(x)=-S(x)$ for all $x \in \R$.
+		\item $C(0)=1,\ C'(0)=0$, and $S(0)=0,\ S'(0)=1$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{corollary}
+	If $C,\ S$ are the functions in \textit{Theorem 8.4.1}, then $C'(x)=-S(x)$ and $S'(x)=C(x)$ for all $x \in \R$. Moreover, these functions have derivatives of all orders.
+\end{corollary}
+
+\begin{corollary}
+	The functions $C$ and $S$ satisfy the Pythagorean Identity:
+	\[(C(x))^2+(S(x))^2=1\ \ \text{for}\ \ x \in \R\]
+\end{corollary}
+
+\begin{theorem}
+	The functions $C$ and $S$ satisfying properties (1) and (2) of \textit{Theorem 8.4.1} are unique.
+\end{theorem}
+
+\begin{definition}
+	The unique functions $C:\R\to\R$ and $S:\R\to\R$ such that $C''(x)=C(x)$ and $S''(x)=-S(x)$ for all $x \in \R$ and $C(0)=1,\ C'(0)=0$ and $S(0)=0,\ S'(0)=1$ are called the \textbf{cosine function} and the \textbf{sine function}, respectively. We ordinarily write
+	\[\cos x := C(x)\ \ \text{and}\ \ \sin x := \S(x)\ \ \text{for}\ \ x \in \R\]
+\end{definition}
+
+\begin{theorem}
+	If $f:\R\to\R$ is such that
+	\[f''(x)=-f(x)\ \ \text{for}\ \ x \in \R\]
+	then there exist real numbers $\alpha,\ \beta$ such that
+	\[f(x)=\alpha C(x)+\beta S(x)\ \ \text{for}\ \ x \in \R\]
+\end{theorem}
+
+\begin{theorem}
+	The function $C$ is even and $S$ is odd in the sense that
+	\begin{enumerate}
+		\item $C(-x)=C(x)$ and $S(-x)=-S(x)$ for $x \in \R$. If $x,\ y \in \R$, then we have the ``addition formulas".
+		\item $C(x+y)=C(x)C(y)-S(x)S(y)$, $S(x+y)=S(x)C(y)+C(x)S(y)$
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	If $x \in \R,\ x \geq 0$, then we have
+	\begin{enumerate}
+		\item $-x \leq S(x) \leq x;$
+		\item $1-\frac{1}{2}x^2 \leq C(x) \leq 1;$
+		\item $x-\frac{1}{6}x^3 \leq S(x) \leq x;$
+		\item $1-\frac{1}{2}x^2 \leq C(x) \leq 1-\frac{1}{2}x^2+\frac{1}{24}x^4$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{lemma}
+	There exists a root $\gamma$ of the cosine function in the interval $(\sqrt{2}, \sqrt{3})$. Moreover $C(x) > 0$ for $x \in [0, \gamma)$. The number $2\gamma$ is the smallest positive root of $S$.
+\end{lemma}
+
+\begin{definition}
+	Let $\pi:=2\gamma$ denote the smallest positive root of $S$.
+\end{definition}
+
+\begin{theorem}
+	The functions $C$ and $S$ have period $2\pi$ in the sense that
+	\begin{enumerate}
+		\item $C(x+2\pi)=C(x)$ and $S(x+2\pi) = S(x)$ for $x \in \R$.
+		      \\Moreover we have
+		\item $S(x)=C(\frac{1}{2}\pi - x) = -C(x+\frac{1}{2}\pi)$, $C(x)=S(\frac{1}{2}\pi-x)=S(x+\frac{1}{2}\pi)$ for all $x \in \R$.
+	\end{enumerate}
+\end{theorem}

chapter-9/absolute-convergence.tex

@@ -0,0 +1,30 @@
+\section{Absolute Convergence}
+
+\begin{definition}
+	Let $X:=(x_n)$ be a sequence in $\R$. We say that the series $\sum x_n$ is \textbf{absolutely convergent} if the series $\sum |x_n|$ is convergent in $\R$. A series is said to be \textbf{conditionally} ( or \textbf{nonabsolutely}) \textbf{convergent} if it is convergent, but it is not absolutely convergent.
+\end{definition}
+
+\begin{theorem}
+	If a series in $\R$ is absolutely convergent, then it is convergent.
+\end{theorem}
+
+\begin{theorem}
+	If a series $\sum x_n$ is convergent, then any series obtained from it by grouping  the terms is also convergent and to the same value.
+\end{theorem}
+
+\begin{definition}
+	A series $\sum y_k$ in $\R$ is a \textbf{rearrangement} of a series $\sum x_n$ if there is a bijection $f$ of $\N$ onto $\N$ such that $y_k=x_{f(k)}$ for all $k \in \N$.
+\end{definition}
+
+\begin{theorem}[\textbf{Rearrangement Theorem}]
+	Let $\sum x_n$ be an absolutely convergent series in $\R$. Then any rearrangement $\sum y_k$ of $\sum x_n$ converges to the same value.
+\end{theorem}
+
+\begin{theorem}
+	If $\sum a_n$ is conditionally convergent, then there exists a rearrangement of $\sum a_n$ such that
+	\begin{enumerate}
+		\item The rearrangement converges to any real number $\alpha$
+		\item The rearrangement diverges to $\pm \infty$
+		\item The rearrangement oscillates between any two real numbers.
+	\end{enumerate}
+\end{theorem}

chapter-9/chapter-9.tex

@@ -0,0 +1,5 @@
+\chapter{Infinite Series}
+\subimport{./}{absolute-convergence.tex}
+\subimport{./}{tests-for-absolute-convergence.tex}
+\subimport{./}{tests-for-nonabsolute-convergence.tex}
+\subimport{./}{series-of-functions.tex}

chapter-9/series-of-functions.tex

@@ -0,0 +1,100 @@
+\section{Series of Functions}
+
+\begin{definition}
+	If $(f_n)$ is a sequence of functions defined on a subset $D$ of $\R$ with values in $\R$, the sequence of \textbf{partial sums} $(s_n)$ of the infinite series $\sum f_n$ is defined for $x$ in $D$ by
+	\begin{align*}
+		s_1(x)          & :=f_1(x)                       \\
+		s_2(x)          & :=s_1(x)+f_2(x)                \\
+		\dots\dots\dots & \dots\dots\dots\dots\dots\dots \\
+		s_{n+1}(x)      & :=s_n(x)+f_{n+1}(x)            \\
+		\dots\dots\dots & \dots\dots\dots\dots\dots\dots
+	\end{align*}
+	In case the sequence $(s_n)$ of functions converges on $D$ to a function $f$, we say that the infinite series of functions $\sum f_n$ \textbf{converges} to $f$ on $D$. We will often write
+	\[\sum f_n\ \ \text{or}\ \ \sum\limits_{n=1}^{\infty}f_n\]
+	to denote either the series or the limit function, when it exists.
+	\\\\If the series $\sum |f_n(x)|$ converges for each $x$ in $D$, we say that $\sum f_n$ is \textbf{absolutely convergent} on $D$. If the sequence $(s_n)$ of partial sums is uniformly convergent on $D$ to $f$, we say that $\sum f_n$ is \textbf{uniformly convergent} on $D$, or that it \textbf{converges to $f$ uniformly on $D$}.
+\end{definition}
+
+\begin{theorem}
+	If $f_n$ is continuous on $D \subseteq \R$ to $\R$ for each $n \in \N$ and if $\sum f_n$ converges to $f$ uniformly on $D$, then $f$ is continuous on $D$.
+\end{theorem}
+
+\begin{theorem}
+	Suppose that the real-valued functions $f_n,\ n \in \N$ are Riemann integrable on the interval $J:=[a,b]$. If the series $\sum f_n$ converges to $f$ uniformly on $J$, then $f$ is Riemann integrable and
+	\[\displaystyle\int_{a}^{b}f=\sum\limits_{n=1}^{\infty}\displaystyle\int_{a}^{b}f_n\]
+\end{theorem}
+
+\begin{theorem}
+	For each $n \in \N$, let $f_n$ be a real-valued function on $J:=[a,b]$ that has a derivative $f'_n$ on $J$. Suppose that the series $\sum f_n$ converges for at least one point of $J$ and that the series of derivatives $\sum f'_n$ converges uniformly on $J$.
+	\\Then there exists a real-valued function $f$ on $J$ such that $\sum f_n$ converges uniformly on $J$ to $f$. In addition, $f$ has a derivative on $J$ and $f'=\sum f'_n$.
+\end{theorem}
+
+\begin{theorem}[\textbf{Cauchy Criterion}]
+	Let $(f_n)$ be a sequence of functions on $D\subseteq \R$ to $\R$. The series $\sum f_n$ is uniformly convergent on $D$ if and only if for every $\varepsilon >0$ there exists an $M(\varepsilon)$ such that if $m>n\geq M(\varepsilon)$, then
+	\[|f_{n+1}(x)+\dots+f_m(x)|<\varepsilon\ \forall\ x \in D\]
+\end{theorem}
+
+\begin{theorem}[\textbf{Weierstrass M-Test}]
+	Let $(M_n)$ be a sequence of positive real numbers such that $|f_n(x)|\leq M_n$ for $x \in D$, $n \in \N$. If the series $\sum M_n$ is convergent, then $\sum f_n$ is uniformly convergent on $D$, $\sum |f_n|$ is uniformly convergent on $D$, and $\sum f_n$ is absolutely convergent on $D$.
+\end{theorem}
+
+\begin{definition}
+	A series of real functions $\sum f_n$ is said to be a \textbf{power series around $x=c$} if the function $f_n$ has the form
+	\[f_n(x)=a_n(x-c)^n\]
+	where $a_n$ and $c$ belong to $\R$ and where $n=0,1,2,\dots$.
+\end{definition}
+
+\begin{definition}
+	Let $\sum a_nx^n$ be a power series. If the sequence $(|a_n|^{1/n})$ is bounded, we set $\rho := \lim\sup(|a_n|^{1/n});$ if this sequence is not bounded we set $\rho=+\infty$. We define the \textbf{radius of convergence} of $\sum a_nx^n$ to be given by
+	\[R:=\begin{cases}
+			0       & \text{if } \rho = +\infty \\
+			1/\rho  & \text{if } 0<\rho<+\infty \\
+			+\infty & \text{if } \rho=0
+		\end{cases}\]
+	The \textbf{interval of convergence} is the open interval $(-R, R)$.
+\end{definition}
+
+\begin{theorem}[\textbf{Cauchy-Hadamard Theorem}]
+	If $R$ is the radius of convergence of the power series $\sum a_nx^n$, then the series is absolutely convergent if $|x|<R$ and is convergent if $|x|>R$.
+\end{theorem}
+
+\begin{remark}
+	It will be noted that the Cauchy-Hadamard Theorem makes no statement as to whether the power series converges when $|x|=R$. Indeed, anything can happen, as the examples
+	\[\sum x^n,\ \sum \frac{1}{n}x^n,\ \sum \frac{1}{n^2}x^n\]
+	show. Since $\lim(n^{1/n})=1$, each of these power series has a radius of convergence equal to 1. The first power series converges at neither of the points $x=-1$ and $x=+1$; the second series converges at $x=-1$ but diverges at $x=+1$; and the third power series converges at both $x=-1$ and $x=+1$. (Find a power series with $R=1$ that converges at $x=+1$ but diverges at $x=-1$.)
+\end{remark}
+
+\begin{theorem}
+	Let $R$ be the radius of convergence of $\sum a_nx^n$ and let $K$ be a closed and bounded interval contained in the interval of convergence $(-R, R)$. Then the power series converges uniformly on $K$.
+\end{theorem}
+
+\begin{theorem}
+	Let $\sum a_n(x-c)^n$ be a power series. Then either
+	\begin{enumerate}
+		\item The series is absolutely convergent on $\R$
+		\item The series converges only at one point, $x=c$
+		\item There exists $R \in \R$ such that $\sum a_n(x-c)^n$ is absolutely convergent for all $|x-c|<R$, and is divergent for all $|x-c|>R$. (Note that the endpoints must be tested separately.)
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+	The limit of a power series is continuous on the interval of convergence. A power series can be integrated term-by-term over any closed and bounded interval contained in the interval of convergence.
+\end{theorem}
+
+\begin{theorem}[\textbf{Differentiation Theorem}]
+	A power series can be differentiated term-by-term within the interval of convergence. In fact, if
+	\[f(x)=\sum\limits_{n=0}^{\infty}a_nx^n,\ \ \text{  then  }\ \ f'(x)=\sum\limits_{n=1}^{\infty}na_nx^{n-1}\ \ \text{  for  }\ \ |x|<R.\]
+	Both series have the same radius of convergence.
+\end{theorem}
+
+\begin{remark}
+	It is to be observed that the theorem makes no assertion about the endpoints of the interval of convergence. If a series is convergent at an endpoint, then the differentiated series may or may not be convergent at this point.
+\end{remark}
+
+\begin{theorem}[\textbf{Uniqueness Theorem}]
+	If $\sum a_nx^n$ and $\sum b_nx^n$ converge on some interval $(-r,r)$, $r>0$, to the same function $f$, then
+	\[a_n=b_n\ \forall\ n \in \N\]
+\end{theorem}
+
+The Taylor Series is
+\[f(x)=\sum\limits_{n=0}^{\infty}\frac{f^{(n)}(c)}{n!}(x-c)^n\]

chapter-9/tests-for-absolute-convergence.tex

@@ -0,0 +1,72 @@
+\section{Tests for Absolute Convergence}
+
+\begin{theorem}[\textbf{Limit Comparison Test, II}]
+	Suppose that $X:=(x_n)$ and $Y:=(y_n)$ are nonzero real sequences and suppose that the following limit exists in $\R$:
+	\[r:=\lim\abs{\frac{x_n}{y_n}}\]
+	\begin{enumerate}
+		\item If $r \neq 0$, then $\sum x_n$ is absolutely convergent if and only if $\sum y_n$ is absolutely convergent.
+		\item If $r=0$ and if $\sum y_n$ is absolutely convergent, then $\sum x_n$ is absolutely convergent.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}[\textbf{Root Test}]
+	Let $X:=(x_n)$ be a sequence in $\R$.
+	\begin{enumerate}
+		\item If there exist $r \in \R$ with $r<1$ and $K \in \N$ such that
+		      \[|x_n|^{1/n}\leq r\ \ \text{for}\ \ n \geq K,\]
+		      then the series $\sum x_n$ is absolutely convergent.
+		\item If there exists $K \in \N$ such that
+		      \[|x_n|^{1/n} \geq 1\ \ \text{for}\ \ n \geq K,\]
+		      then the series $\sum x_n$ is divergent.
+	\end{enumerate}
+\end{theorem}
+
+\begin{corollary}
+	Let $X:=(x_n)$ be a sequence in $\R$ and suppose that the limit
+	\[r:=\lim |x_n|^{1/n}\]
+	exists in $\R$. Then $\sum x_n$ is absolutely convergent when $r < 1$ and is divergent when $r > 1$.
+\end{corollary}
+
+\begin{theorem}[\textbf{Ratio Test}]
+	Let $X := (x_n)$ be a sequence of nonzero real numbers.
+	\begin{enumerate}
+		\item If there exist $r \in \R$ with $0<r<1$ and $K \in \N$ such that
+		      \[\abs{\frac{x_{n+1}}{x_n}}\leq r\ \ \text{for}\ \ n \geq K,\]
+		      then the series $\sum x_n$ is absolutely convergent.
+		\item If there exists $K \in \N$ such that
+		      \[\abs{\frac{x_{n+1}}{x_n}}\geq 1\ \ \text{for}\ \ n \geq K,\]
+		      then the series $\sum x_n$ is divergent.
+	\end{enumerate}
+\end{theorem}
+
+\begin{corollary}
+	Let $X :=(x_n)$ be a nonzero sequence in $\R$ and suppose that the limit
+	\[r:=\lim\abs{\frac{x_{n+1}}{x_n}}\]
+	exists in $\R$. Then $\sum x_n$ is absolutely convergent when $r<1$ and is divergent when $r>1$.
+\end{corollary}
+
+\begin{theorem}[\textbf{Integral Test}]
+	Let $f$ be a positive, decreasing function on $\{t:t\geq 1\}$. Then the series $\sum\limits_{k=1}^{\infty}f(k)$ converges if and only if the improper integral
+	\[\displaystyle\int_{1}^{\infty}f(t)dt=\lim\limits_{b \to \infty}\displaystyle\int_{1}^{b}f(t)dt\]
+	exists. In the case of convergence, the partial sum $s_n=\sum\limits_{k=1}^{n}f(k)$ and the sum $s=\sum\limits_{k=1}^{\infty} f(k)$ satisfy the estimate
+	\[\displaystyle\int_{n+1}^{\infty}f(t)dt\leq s-s_n \leq \displaystyle\int_{n}^{\infty}f(t)dt\]
+\end{theorem}
+
+\begin{theorem}[\textbf{Raabe's Test}]
+	Let $X:=(x_n)$ be a sequence of nonzero real numbers.
+	\begin{enumerate}
+		\item If there exist numbers $a >1$ and $K \in \N$ such that
+		      \[\abs{\frac{x_{n+1}}{x_n}}\leq 1 - \frac{a}{n}\ \ \text{for}\ \ n \geq K,\]
+		      then $\sum x_n$ is absolutely convergent.
+
+		\item If there exist real numbers $a \leq 1$ and $K \in \N$ such that
+		      \[\abs{\frac{x_{n+1}}{x_n}}\geq 1-\frac{a}{n}\ \ \text{for}\ \ n \geq K,\]
+		      then $\sum x_n$ is not absolutely convergent.
+	\end{enumerate}
+\end{theorem}
+
+\begin{corollary}
+	Let $X:=(x_n)$ be a nonzero sequence in $\R$ and let
+	\[a:=\lim \left(n \left(1-\abs{\frac{x_{n+1}}{x_n}}\right)\right)\]
+	whenever this limit exists. Then $\sum x_n$ is absolutely convergent when $a > 1$ and is not absolutely convergent when $a <1$.
+\end{corollary}

chapter-9/tests-for-nonabsolute-convergence.tex

@@ -0,0 +1,22 @@
+\section{Tests for Nonabsolute Convergence}
+
+\begin{definition}
+	A sequence $X:=(x_n)$ of nonzero real numbers is said to be \textbf{alternating} if the terms $(-1)^{n+1}$, $n \in \N$, are all positive (or all negative) real numbers. If the sequence $X:=(x_n)$ is alternative, we say that the series $\sum x_n$ it generates is an \textbf{alternating series}.
+\end{definition}
+
+\begin{theorem}[\textbf{Alternating Series Test}]
+	Let $Z:=(z_n)$ be a decreasing sequence of strictly positive numbers with $\lim (z_n)=0$. Then the alternating series $\sum (-1)^{n+1} z_n$ is convergent.
+\end{theorem}
+
+\begin{lemma}[\textbf{Abel's Lemma}]
+	Let $X:=(x_n)$ and $Y:=(y_n)$ be sequences in $\R$ and let the partial sums of $\sum y_n$ be denoted by $(s_n)$ with $s_0:=0$. If $m >n$, then
+	\[\sum\limits_{k=n+1}^{m}x_ky_k=(x_ms_m-x_{n+1}s_n)+\sum\limits_{k=n+1}^{m-1}(x_k-x_{k+1})s_k\]
+\end{lemma}
+
+\begin{theorem}[\textbf{Dirichlet's Test}]
+	If $X:=(x_n)$ is a decreasing sequence with $\lim x_n=0$, and if the partial sums $(s_n)$ of $\sum y_n$ are bounded, then the series $\sum x_ny_n$ is convergent.
+\end{theorem}
+
+\begin{theorem}[\textbf{Abel's Test}]
+	If $X:=(x_n)$ is a convergent monotone sequence and the series $\sum y_n$ is convergent, then the series $\sum x_ny_n$ is also convergent.
+\end{theorem}

init.sty

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+
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