Let $A\subset\mathbb{R}$ a nonempty set of real numbers bounded above and $u$ be an upper bound of $A$. Prove that if $u\in A$, then $u=\sup A$.
Proving that if $u \in A$ is an upper bound of $A$, then $u = \sup A$
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analysis
metric-spaces
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1 Answers
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Recall the definition of supremum: If the non-empty set of real numbers $S$ is bounded above then $u$ is the supremum of $S$ if both the following hold:
$\ \ \ $1) $u$ is an upper bound of $S$
and
$\ \ \ $2) if $v$ is any upper bound of $S$, then $u\le v$.
A hint for your problem: Suppose $v$ is another upper bound of $A$. Think about condition 2), keeping in mind that $u\in A$.