From 2c28016505d12771039f4171c599bf86675c7669 Mon Sep 17 00:00:00 2001
From: gapry <gapry@protonmail.com>
Date: Sun, 14 Jan 2024 23:12:47 +0800
Subject: [PATCH] BabyRudin: Solution: Ex05 in Ch01

---
 Books/BabyRudin/Chapter01/ex05.tex | 14 ++++++++++++++
 1 file changed, 14 insertions(+)
 create mode 100644 Books/BabyRudin/Chapter01/ex05.tex

diff --git a/Books/BabyRudin/Chapter01/ex05.tex b/Books/BabyRudin/Chapter01/ex05.tex
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+\subsection*{Exercise 05}
+\begin{flushleft}
+$\text{(1) } \exists\ \alpha \in R \ni  x \geq  \alpha \ \forall  x \in  A$ \\
+$\text{(2) } \exists\ \alpha \in R \ni -x \leq -\alpha \ \forall -x \in -A$ \\
+\end{flushleft}
+
+\begin{flushleft}
+(1) $\implies  \alpha = \inf( A)$ \\
+(2) $\implies -\alpha = \sup(-A)$
+\end{flushleft}
+
+\begin{flushleft}
+Obviously, $a = -(-a) \implies \inf(A) = -\sup(-A)$
+\end{flushleft}