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Math Help - Compact Sets

  1. #1
    Junior Member
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    Wink Compact Sets

    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!
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  2. #2
    Member
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    Jun 2009
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    Use equicontinuity of B (Arzelą-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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