Compact Sets

**dabien** · December 9th 2009, 08:01 PM

Let $A\subset R^n$ be compact and let $B\subset$ C(A,R) be compact. Show that there are an $f_0\in B$ and an $x_0\in A$ such that $g(x)\leq f_0(x_0)$ for all $g\in B$ and $x\in A$

Help please!

**Enrique2** · December 10th 2009, 10:54 AM

Use equicontinuity of B (Arzelà-Ascoli) to show that

$G:A\times B \to \mathbb{R}$ , $(x,f)\mapsto f(x)$ is continuous. Then the statement follows from the fact that $A\times B$ is compact.

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