Proof verification: If $a\in A$ is an upper bound for $A$, then $a=\sup A$

by csch2   Last Updated June 12, 2019 08:20 AM

Prove that if $a$ is an upper bound for $A$, and if $a$ is also an element of $A$, then it must be that $a=\sup A$.

We are given the following lemma:

Lemma 1.3.8: Assume $s\in\textbf{R}$ is an upper bound for a set $A\subseteq\textbf{R}$. Then, $s=\sup A$ if and only if, for every choice of $\epsilon>0$, there exists an element $a\in A$ satisfying $s-a<\epsilon$.

Proof: It is given that $a$ is an upper bound for $A$. To verify that $a=\sup A$, we use Lemma 1.3.8. We want to show that $a-\epsilon<a_0$ for some $a_0\in A,\epsilon>0$. Since $a\in A$, let $a=a_0$. Then we get that $a-\epsilon<a$, which holds for all $\epsilon>0$. Therefore, by Lemma 1.3.8, we have that $a=\sup A$.

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