commit d19e454f2765ab0d84910279b3b3a95d34c098cf Author: Alex Clarke Date: Tue Jan 9 13:01:34 2024 -0700 Created the Real Analysis Theorems and Definitions packet diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..7f7bed5 --- /dev/null +++ b/.gitignore @@ -0,0 +1,303 @@ +## 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 + diff --git a/README.md b/README.md new file mode 100644 index 0000000..6bea290 --- /dev/null +++ b/README.md @@ -0,0 +1,20 @@ +# 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 diff --git a/Real-Analysis-Theorems-and-Definitions.pdf b/Real-Analysis-Theorems-and-Definitions.pdf new file mode 100644 index 0000000..549ec90 Binary files /dev/null and b/Real-Analysis-Theorems-and-Definitions.pdf differ diff --git a/Real-Analysis-Theorems-and-Definitions.tex b/Real-Analysis-Theorems-and-Definitions.tex new file mode 100644 index 0000000..a15a0b0 --- /dev/null +++ b/Real-Analysis-Theorems-and-Definitions.tex @@ -0,0 +1,32 @@ +\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} diff --git a/chapter-1/chapter-1.tex b/chapter-1/chapter-1.tex new file mode 100644 index 0000000..4b2a9c4 --- /dev/null +++ b/chapter-1/chapter-1.tex @@ -0,0 +1,4 @@ +\chapter{Preliminaries} +\subimport{./}{sets-and-functions.tex} +\subimport{./}{mathematical-induction.tex} +\subimport{./}{finite-and-infinite-sets.tex} diff --git a/chapter-1/finite-and-infinite-sets.tex b/chapter-1/finite-and-infinite-sets.tex new file mode 100644 index 0000000..cbbb8b6 --- /dev/null +++ b/chapter-1/finite-and-infinite-sets.tex @@ -0,0 +1,89 @@ +\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} diff --git a/chapter-1/mathematical-induction.tex b/chapter-1/mathematical-induction.tex new file mode 100644 index 0000000..03f6e2f --- /dev/null +++ b/chapter-1/mathematical-induction.tex @@ -0,0 +1,35 @@ +\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} diff --git a/chapter-1/sets-and-functions.tex b/chapter-1/sets-and-functions.tex new file mode 100644 index 0000000..faf2ba3 --- /dev/null +++ b/chapter-1/sets-and-functions.tex @@ -0,0 +1,77 @@ +\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} diff --git a/chapter-10/chapter-10.tex b/chapter-10/chapter-10.tex new file mode 100644 index 0000000..17ecca1 --- /dev/null +++ b/chapter-10/chapter-10.tex @@ -0,0 +1,5 @@ +\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} diff --git a/chapter-10/convergence-theorems.tex b/chapter-10/convergence-theorems.tex new file mode 100644 index 0000000..023bc59 --- /dev/null +++ b/chapter-10/convergence-theorems.tex @@ -0,0 +1,64 @@ +\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} diff --git a/chapter-10/definition-and-main-properties.tex b/chapter-10/definition-and-main-properties.tex new file mode 100644 index 0000000..35f413c --- /dev/null +++ b/chapter-10/definition-and-main-properties.tex @@ -0,0 +1,96 @@ +\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} diff --git a/chapter-10/improper-and-lebesgue-integrals.tex b/chapter-10/improper-and-lebesgue-integrals.tex new file mode 100644 index 0000000..68302b7 --- /dev/null +++ b/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} diff --git a/chapter-10/infinite-intervals.tex b/chapter-10/infinite-intervals.tex new file mode 100644 index 0000000..a67c7d8 --- /dev/null +++ b/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} diff --git a/chapter-11/chapter-11.tex b/chapter-11/chapter-11.tex new file mode 100644 index 0000000..c870091 --- /dev/null +++ b/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} diff --git a/chapter-11/compact-sets.tex b/chapter-11/compact-sets.tex new file mode 100644 index 0000000..fd95726 --- /dev/null +++ b/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} diff --git a/chapter-11/continuous-functions.tex b/chapter-11/continuous-functions.tex new file mode 100644 index 0000000..3e3d9fd --- /dev/null +++ b/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} diff --git a/chapter-11/metric-spaces.tex b/chapter-11/metric-spaces.tex new file mode 100644 index 0000000..2f370ed --- /dev/null +++ b/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} diff --git a/chapter-11/open-and-closed-sets-in-R.tex b/chapter-11/open-and-closed-sets-in-R.tex new file mode 100644 index 0000000..115c30b --- /dev/null +++ b/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} diff --git a/chapter-2/absolute-value-and-the-real-line.tex b/chapter-2/absolute-value-and-the-real-line.tex new file mode 100644 index 0000000..6a544ca --- /dev/null +++ b/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} diff --git a/chapter-2/applications-of-the-supremum-property.tex b/chapter-2/applications-of-the-supremum-property.tex new file mode 100644 index 0000000..05d0c96 --- /dev/null +++ b/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 $xb$ 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 $ca0$. + \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} diff --git a/chapter-2/the-completeness-property-of-R.tex b/chapter-2/the-completeness-property-of-R.tex new file mode 100644 index 0000000..f5951f4 --- /dev/null +++ b/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} diff --git a/chapter-3/chapter-3.tex b/chapter-3/chapter-3.tex new file mode 100644 index 0000000..d76227e --- /dev/null +++ b/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} diff --git a/chapter-3/introduction-to-infinite-series.tex b/chapter-3/introduction-to-infinite-series.tex new file mode 100644 index 0000000..d589924 --- /dev/null +++ b/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} diff --git a/chapter-3/limit-theorems.tex b/chapter-3/limit-theorems.tex new file mode 100644 index 0000000..d5f2f66 --- /dev/null +++ b/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} diff --git a/chapter-3/monotone-sequences.tex b/chapter-3/monotone-sequences.tex new file mode 100644 index 0000000..50c3097 --- /dev/null +++ b/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} diff --git a/chapter-3/properly-divergent-sequences.tex b/chapter-3/properly-divergent-sequences.tex new file mode 100644 index 0000000..c8defcc --- /dev/null +++ b/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} diff --git a/chapter-3/sequences-and-their-limits.tex b/chapter-3/sequences-and-their-limits.tex new file mode 100644 index 0000000..844d9ea --- /dev/null +++ b/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} diff --git a/chapter-3/subsequences-and-the-bolzano-wierstrass-theorem.tex b/chapter-3/subsequences-and-the-bolzano-wierstrass-theorem.tex new file mode 100644 index 0000000..9c85566 --- /dev/null +++ b/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} diff --git a/chapter-3/the-cauchy-criterion.tex b/chapter-3/the-cauchy-criterion.tex new file mode 100644 index 0000000..1e3c114 --- /dev/null +++ b/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 $C$, $0 0$ 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} diff --git a/chapter-4/limits-of-functions.tex b/chapter-4/limits-of-functions.tex new file mode 100644 index 0000000..114cb2d --- /dev/null +++ b/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 +\[\sign (x):=\begin{cases} + +1 & \text{for } x>0, \\ + 0 & \text{for } x=0, \\ + -1 & \text{for } x < 0. + \end{cases}\] diff --git a/chapter-4/some-extensions-of-the-limit-concept.tex b/chapter-4/some-extensions-of-the-limit-concept.tex new file mode 100644 index 0000000..5fa0902 --- /dev/null +++ b/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 $A \cap (-\infty, c)=\{x \in A: x 0$ 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} diff --git a/chapter-5/chapter-5.tex b/chapter-5/chapter-5.tex new file mode 100644 index 0000000..32fc9b6 --- /dev/null +++ b/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} diff --git a/chapter-5/combinations-of-continuous-functions.tex b/chapter-5/combinations-of-continuous-functions.tex new file mode 100644 index 0000000..5f91a8a --- /dev/null +++ b/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} diff --git a/chapter-5/continuity-and-gauges.tex b/chapter-5/continuity-and-gauges.tex new file mode 100644 index 0000000..a3f9ee1 --- /dev/null +++ b/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} diff --git a/chapter-5/continuous-functions-on-intervals.tex b/chapter-5/continuous-functions-on-intervals.tex new file mode 100644 index 0000000..a87930b --- /dev/null +++ b/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 $f(a) < k 0$, 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} diff --git a/chapter-5/monotone-and-inverse-functions.tex b/chapter-5/monotone-and-inverse-functions.tex new file mode 100644 index 0000000..697be84 --- /dev/null +++ b/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 $x_1 g(x_2)$. + \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} diff --git a/chapter-5/uniform-continuity.tex b/chapter-5/uniform-continuity.tex new file mode 100644 index 0000000..e73b4c5 --- /dev/null +++ b/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} diff --git a/chapter-6/chapter-6.tex b/chapter-6/chapter-6.tex new file mode 100644 index 0000000..19d8b69 --- /dev/null +++ b/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} diff --git a/chapter-6/lhopitals-rules.tex b/chapter-6/lhopitals-rules.tex new file mode 100644 index 0000000..ca9738a --- /dev/null +++ b/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} diff --git a/chapter-6/taylors-theorem.tex b/chapter-6/taylors-theorem.tex new file mode 100644 index 0000000..fcf32e9 --- /dev/null +++ b/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} diff --git a/chapter-6/the-derivative.tex b/chapter-6/the-derivative.tex new file mode 100644 index 0000000..6eb6535 --- /dev/null +++ b/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} diff --git a/chapter-6/the-mean-value-theorem.tex b/chapter-6/the-mean-value-theorem.tex new file mode 100644 index 0000000..1dd335f --- /dev/null +++ b/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} diff --git a/chapter-7/approximate-integration.tex b/chapter-7/approximate-integration.tex new file mode 100644 index 0000000..f9ec467 --- /dev/null +++ b/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: +\[a0$ 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 $a=c_00$ 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} diff --git a/chapter-7/the-darboux-integral.tex b/chapter-7/the-darboux-integral.tex new file mode 100644 index 0000000..07efa69 --- /dev/null +++ b/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} diff --git a/chapter-7/the-fundamental-theorem.tex b/chapter-7/the-fundamental-theorem.tex new file mode 100644 index 0000000..280bff3 --- /dev/null +++ b/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} diff --git a/chapter-8/chapter-8.tex b/chapter-8/chapter-8.tex new file mode 100644 index 0000000..7e13c1c --- /dev/null +++ b/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} diff --git a/chapter-8/interchange-of-limits.tex b/chapter-8/interchange-of-limits.tex new file mode 100644 index 0000000..9568ca3 --- /dev/null +++ b/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} diff --git a/chapter-8/pointwise-and-uniform-convergence.tex b/chapter-8/pointwise-and-uniform-convergence.tex new file mode 100644 index 0000000..9d74784 --- /dev/null +++ b/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} diff --git a/chapter-8/the-exponential-and-logarithmic-functions.tex b/chapter-8/the-exponential-and-logarithmic-functions.tex new file mode 100644 index 0000000..24181ac --- /dev/null +++ b/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} diff --git a/chapter-8/the-trigonometric-functions.tex b/chapter-8/the-trigonometric-functions.tex new file mode 100644 index 0000000..b296bbc --- /dev/null +++ b/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} diff --git a/chapter-9/absolute-convergence.tex b/chapter-9/absolute-convergence.tex new file mode 100644 index 0000000..22acce3 --- /dev/null +++ b/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} diff --git a/chapter-9/chapter-9.tex b/chapter-9/chapter-9.tex new file mode 100644 index 0000000..e15637f --- /dev/null +++ b/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} diff --git a/chapter-9/series-of-functions.tex b/chapter-9/series-of-functions.tex new file mode 100644 index 0000000..5cfda1c --- /dev/null +++ b/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 $|x|R$. +\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 $|x-c|R$. (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|0$, to the same function $f$, then + \[a_n=b_n\ \forall\ n \in \N\] +\end{theorem} + +The Taylor Series is +\[f(x)=\sum\limits_{n=0}^{\infty}\frac{f^{(n)}(c)}{n!}(x-c)^n\] diff --git a/chapter-9/tests-for-absolute-convergence.tex b/chapter-9/tests-for-absolute-convergence.tex new file mode 100644 index 0000000..a3de677 --- /dev/null +++ b/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 $01$. +\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} diff --git a/chapter-9/tests-for-nonabsolute-convergence.tex b/chapter-9/tests-for-nonabsolute-convergence.tex new file mode 100644 index 0000000..12db348 --- /dev/null +++ b/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} diff --git a/init.sty b/init.sty new file mode 100644 index 0000000..1e9cc47 --- /dev/null +++ b/init.sty @@ -0,0 +1,57 @@ +\ProvidesPackage{init} + +\usepackage{import} +\usepackage[utf8]{inputenc} +\usepackage{pgfplots} +\usepackage[english]{babel} +\usepackage{amsthm} +\usepackage{thmtools} +\usepackage{hyperref} +\usepackage{cancel} +\usepackage{mathtools} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{graphicx} +\usepackage{relsize} +\usepackage{listings} +\graphicspath{ {./images/} } +\usepackage{array} +\usepackage{tikz} +\usetikzlibrary{arrows} +\usepackage[left=2cm, right=2.5cm, top=2.5cm, bottom=2.5cm]{geometry} +\usepackage{enumitem} +\usepackage{mathrsfs} + +% Math Functions +\newcommand{\limx}[2]{\displaystyle\lim\limits_{#1 \to #2}} +\newcommand{\st}{\text{ s.t. }} +\newcommand{\abs}[1]{\left\lvert #1 \right\rvert} +\newcommand{\dotp}{\dot{\mathcal{P}}} +\newcommand{\dotq}{\dot{\mathcal{Q}}} +\newcommand{\Int}[1]{\text{int}\left(#1\right)} +\newcommand{\cl}[1]{\text{cl}\left(#1\right)} +\newcommand{\bd}[1]{\text{bd}\left(#1\right)} +\newcommand{\lr}[1]{\langle #1 \rangle)} +\newcommand{\lspan}[1]{\text{span}\left(#1\right)} +\newcommand{\ldim}[1]{\text{dim}\left(#1\right)} +\newcommand{\nullity}[1]{\text{nullity}\left(#1\right)} +\newcommand{\rank}[1]{\text{rank}\left(#1\right)} +\newcommand{\ldet}[1]{\text{det}\left(#1\right)} +\newcommand{\ltr}[1]{\text{tr}\left(#1\right)} +\newcommand{\norm}[1]{\left\lVert#1\right\rVert} +\DeclareMathOperator{\sign}{sgn} +\renewcommand{\qedsymbol}{$\blacksquare$} + +% Special Sets +\newcommand{\R}{\mathbb{R}} +\newcommand{\N}{\mathbb{N}} +\newcommand{\Q}{\mathbb{Q}} +\newcommand{\C}{\mathbb{C}} +\newcommand{\Z}{\mathbb{Z}} + +% Theorem Styles +\declaretheorem[numberwithin=section, style=definition]{theorem, definition, notation, lemma, corollary, remark, example} + +% Formatting +\setlist[enumerate]{font=\bfseries}