117 lines
2.6 KiB
TeX
117 lines
2.6 KiB
TeX
\documentclass{article}
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\usepackage[shortlabels]{enumitem}
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\input{preamble}
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\newcommand{\link}[1]{\lean{../..}{Exercises/Apostol/Chapter_I_3}
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{Exercises.Apostol.Chapter\_I\_3.Real.#1}}
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\begin{document}
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\section*{Theorem I.27}%
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\label{sec:theorem-i.27}
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Every nonempty set $S$ that is bounded below has a greatest lower bound; that
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is, there is a real number $L$ such that $L = \inf{S}$.
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\begin{proof}
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\link{exists_isGLB}
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\end{proof}
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\section*{Theorem I.29}%
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\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}
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\link{exists_pnat_geq_self}
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\end{proof}
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\section*{Theorem I.30 (Archimedean Property of the Reals)}%
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\label{sec:theorem-i.30}
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If $x > 0$ and if $y$ is an arbitrary real number, there exists a positive
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integer $n$ such that $nx > y$.
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\begin{proof}
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\link{exists_pnat_mul_self_geq_of_pos}
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\end{proof}
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\section*{Theorem I.31}%
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\label{sec:theorem-i.31}
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If three real numbers $a$, $x$, and $y$ satisfy the inequalities
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$$a \leq x \leq a + \frac{y}{n}$$
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for every integer $n \geq 1$, then $x = a$.
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\begin{proof}
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\link{forall_pnat_leq_self_leq_frac_imp_eq}
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\end{proof}
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\section*{Theorem I.32}%
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\label{sec:theorem-i.32}
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Let $h$ be a given positive number and let $S$ be a set of real numbers.
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\begin{enumerate}[(a)]
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\item If $S$ has a supremum, then for some $x$ in $S$ we have
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$$x > \sup{S} - h.$$
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\item If $S$ has an infimum, then for some $x$ in $S$ we have
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$$x < \inf{S} + h.$$
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\end{enumerate}
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\begin{proof}
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\begin{enumerate}[(a)]
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\item \link{sup_imp_exists_gt_sup_sub_delta}
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\item \link{inf_imp_exists_lt_inf_add_delta}
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\end{enumerate}
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\end{proof}
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\section*{Theorem I.33 (Additive Property)}%
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\label{sec:theorem-i.33}
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Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set
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$$C = \{a + b : a \in A, b \in B\}.$$
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\begin{enumerate}[(a)]
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\item If each of $A$ and $B$ has a supremum, then $C$ has a supremum, and
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$$\sup{C} = \sup{A} + \sup{B}.$$
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\item If each of $A$ and $B$ has an infimum, then $C$ has an infimum, and
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$$\inf{C} = \inf{A} + \inf{B}.$$
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\end{enumerate}
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\begin{proof}
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\begin{enumerate}[(a)]
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\item \link{sup_minkowski_sum_eq_sup_add_sup}
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\item \link{inf_minkowski_sum_eq_inf_add_inf}
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\end{enumerate}
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\end{proof}
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\section*{Theorem I.34}%
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\label{sec:theorem-i.34}
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Given two nonempty subsets $S$ and $T$ of $\mathbb{R}$ such that
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$$s \leq t$$
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for every $s$ in $S$ and every $t$ in $T$. Then $S$ has a supremum, and $T$
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has an infimum, and they satisfy the inequality
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$$\sup{S} \leq \inf{T}.$$
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\begin{proof}
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\link{forall_mem_le_forall_mem_imp_sup_le_inf}
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\end{proof}
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\end{document}
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