I_3_11 Apostol. Tex proof.

Also nest paragraphs in LaTeX structures.
finite-set-exercises
Joshua Potter 2023-04-10 16:25:32 -06:00
parent a2b138233a
commit 418103eb8c
4 changed files with 47 additions and 22 deletions

View File

@ -1,4 +1,5 @@
\documentclass{article} \documentclass{article}
\usepackage[shortlabels]{enumitem}
\input{../../common/preamble} \input{../../common/preamble}
@ -6,8 +7,8 @@
\begin{xtheorem}{I.27} \begin{xtheorem}{I.27}
Every nonempty set $S$ that is bounded below has a greatest lower bound; Every nonempty set $S$ that is bounded below has a greatest lower bound; that
that is, there is a real number $L$ such that $L = \inf{S}$. is, there is a real number $L$ such that $L = \inf{S}$.
\end{xtheorem} \end{xtheorem}
@ -31,7 +32,8 @@ For every real $x$ there exists a positive integer $n$ such that $n > x$.
\begin{xtheorem}{I.30}[Archimedean Property of the Reals] \begin{xtheorem}{I.30}[Archimedean Property of the Reals]
If $x > 0$ and if $y$ is an arbitrary real number, there exists a positive integer $n$ such that $nx > y$. If $x > 0$ and if $y$ is an arbitrary real number, there exists a positive
integer $n$ such that $nx > y$.
\end{xtheorem} \end{xtheorem}
@ -55,4 +57,27 @@ for every integer $n \geq 1$, then $x = a$.
\end{proof} \end{proof}
\begin{xtheorem}{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}
\end{xtheorem}
\begin{proof}
\ % Force space prior to *Proof.*
\begin{enumerate}[(a)]
\item \href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.arb_close_to_sup}
\item \href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.arb_close_to_inf}
\end{enumerate}
\end{proof}
\end{document} \end{document}