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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- Mar 24th 2007, 09:47 PMtttcomraderCompact 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.

- Mar 24th 2007, 10:37 PMecMathGeek
- Mar 24th 2007, 10:48 PMJhevon
- Mar 25th 2007, 04:50 AMPlato
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.)