bookshelf/Exercises/Apostol/Chapter_I_3.tex

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\documentclass{article}
\usepackage[shortlabels]{enumitem}
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\input{preamble}
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\newcommand{\link}[1]{\href{../../../../Exercises/Apostol/Chapter_I_3.html\##1}{#1}}
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\begin{document}
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\section*{Theorem I.27}%
\label{sec:theorem-i.27}
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Every nonempty set $S$ that is bounded below has a greatest lower bound; that
is, there is a real number $L$ such that $L = \inf{S}$.
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\begin{proof}
\link{Real.exists\_isGLB}
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\end{proof}
\section*{Theorem I.29}%
\label{sec:theorem-i.29}
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For every real $x$ there exists a positive integer $n$ such that $n > x$.
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\begin{proof}
\link{Real.exists\_pnat\_geq\_self}
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\end{proof}
\section*{Theorem I.30 (Archimedean Property of the Reals)}%
\label{sec:theorem-i.30}
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If $x > 0$ and if $y$ is an arbitrary real number, there exists a positive
integer $n$ such that $nx > y$.
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\begin{proof}
\link{Real.exists\_pnat\_mul\_self\_geq\_of\_pos}
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\end{proof}
\section*{Theorem I.31}%
\label{sec:theorem-i.31}
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If three real numbers $a$, $x$, and $y$ satisfy the inequalities
$$a \leq x \leq a + \frac{y}{n}$$
for every integer $n \geq 1$, then $x = a$.
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\begin{proof}
\link{Real.forall\_pnat\_leq\_self\_leq\_frac\_imp\_eq}
\end{proof}
\section*{Theorem I.32}%
\label{sec:theorem-i.32}
Let $h$ be a given positive number and let $S$ be a set of real numbers.
\begin{enumerate}[(a)]
\item If $S$ has a supremum, then for some $x$ in $S$ we have
$$x > \sup{S} - h.$$
\item If $S$ has an infimum, then for some $x$ in $S$ we have
$$x < \inf{S} + h.$$
\end{enumerate}
\begin{proof}
\begin{enumerate}[(a)]
\item \link{Real.sup\_imp\_exists\_gt\_sup\_sub\_delta}
\item \link{Real.inf\_imp\_exists\_lt\_inf\_add\_delta}
\end{enumerate}
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\end{proof}
\section*{Theorem I.33 (Additive Property)}%
\label{sec:theorem-i.33}
Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set
$$C = \{a + b : a \in A, b \in B\}.$$
\begin{enumerate}[(a)]
\item If each of $A$ and $B$ has a supremum, then $C$ has a supremum, and
$$\sup{C} = \sup{A} + \sup{B}.$$
\item If each of $A$ and $B$ has an infimum, then $C$ has an infimum, and
$$\inf{C} = \inf{A} + \inf{B}.$$
\end{enumerate}
\begin{proof}
\begin{enumerate}[(a)]
\item \link{Real.sup\_minkowski\_sum\_eq\_sup\_add\_sup}
\item \link{Real.inf\_minkowski\_sum\_eq\_inf\_add\_inf}
\end{enumerate}
\end{proof}
\section*{Theorem I.34}%
\label{sec:theorem-i.34}
Given two nonempty subsets $S$ and $T$ of $\mathbb{R}$ such that
$$s \leq t$$
for every $s$ in $S$ and every $t$ in $T$. Then $S$ has a supremum, and $T$
has an infimum, and they satisfy the inequality
$$\sup{S} \leq \inf{T}.$$
\begin{proof}
\link{Real.forall\_mem\_le\_forall\_mem\_imp\_sup\_le\_inf}
\end{proof}
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\end{document}