125 lines
2.9 KiB
TeX
125 lines
2.9 KiB
TeX
\documentclass{article}
|
|
\usepackage[shortlabels]{enumitem}
|
|
|
|
\input{../../../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.sup_imp_exists_gt_sup_sub_delta}
|
|
\item \href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.inf_imp_exists_lt_inf_add_delta}
|
|
\end{enumerate}
|
|
|
|
\end{proof}
|
|
|
|
\begin{xtheorem}{I.33}[Additive Property]
|
|
|
|
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}
|
|
|
|
\end{xtheorem}
|
|
|
|
\begin{proof}
|
|
|
|
\ % Force space prior to *Proof.*
|
|
|
|
\begin{enumerate}[(a)]
|
|
\item \href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.sup_minkowski_sum_eq_sup_add_sup}
|
|
\item \href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.inf_minkowski_sum_eq_inf_add_inf}
|
|
\end{enumerate}
|
|
|
|
\end{proof}
|
|
|
|
\begin{xtheorem}{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
|
|
$$\sup{S} \leq \inf{T}.$$
|
|
|
|
\end{xtheorem}
|
|
|
|
\begin{proof}
|
|
|
|
\href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.forall_mem_le_forall_mem_imp_sup_le_inf}
|
|
|
|
\end{proof}
|
|
|
|
\end{document}
|