bookshelf/one-variable-calculus/Apostol/Chapter_I_3.tex

\documentclass{article}
\usepackage[shortlabels]{enumitem}

\input{../../common/preamble}

\begin{document}

\begin{xtheorem}{I.27}

  Every nonempty set $S$ that is bounded below has a greatest lower bound; that
  is, there is a real number $L$ such that $L = \inf{S}$.

\end{xtheorem}

\begin{proof}

  \href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.exists_isGLB}

\end{proof}

\begin{xtheorem}{I.29}

  For every real $x$ there exists a positive integer $n$ such that $n > x$.

\end{xtheorem}

\begin{proof}

  \href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.exists_pnat_geq_self}

\end{proof}

\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 $nx > y$.

\end{xtheorem}

\begin{proof}

  \href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.exists_pnat_mul_self_geq_of_pos}

\end{proof}

\begin{xtheorem}{I.31}

  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 $x = a$.

\end{xtheorem}

\begin{proof}

  \href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.forall_pnat_leq_self_leq_frac_imp_eq}

\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}