\documentclass{article}

\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}

\end{document}