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Math Help - Compact and Close Set

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    Compact and Close Set

    Let K and F be nonempty subsets of R^n. Suppose that K is compact and F is closed. Define d = inf {||x-y|| : x in K, y in F}. Prove that d = 0 if and only if K intersect F = nonempty set.
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    Senior Member ecMathGeek's Avatar
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    Quote Originally Posted by tttcomrader View Post
    Let K and F be nonempty subsets of R^n. Suppose that K is compact and F is closed. Define d = inf {||x-y|| : x in K, y in F}. Prove that d = 0 if and only if K intersect F = nonempty set.
    This is Calculus?
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by ecMathGeek View Post
    This is Calculus?
    it seems to be some number theory course, or maybe abstract/linear algebra
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    Suppose that z belongs to both F and K. Then ||z-z||=0 thus d=0.

    Now if d=0 then for each positive integer n, there are points in K, k_n, and f_n in F such that ||k_n f_n||<(1/n).
    By the compactness of K the sequence (k_n) converges to a point in K, k, but k is a limit point of F and because F is closed k belongs to F.
    (You may have to be more careful picking the points, but that is the general idea.)
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