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