# Let $A$ be a nonempty bounded subset of $\mathbb R$ and let $B$ be the set of all upper bounds for $A$. Prove that $\sup A=\inf B$

Let $A$ be a nonempty bounded subset of $\mathbb R$ and let $B$ be the set of all upper bounds for $A$. Prove that $\sup A= \inf B$. Can someone please help me? I'm very confused as to what to do.

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How do you define $\sup A$? –  Antonio Vargas Sep 18 '12 at 18:33

Well, by definition of supremum, $\sup A\in B$ and for all $x\in B$, $\sup A\leq x$. Then $\sup A=\min B=\inf B$.