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发帖数: 10 | 1 请大侠指点，多谢！
Let (X, d) be a metric space. For nonempty subsets E, F in X, the distance
between E and F is defined to be dist(E,F) = inf{d(x,y): x in E, y in F}
Suppose E and F are disjoint closed subsets of X, E is compact, and X = R^n,
prove that there exist x in E and y in F such that d(x,y) = dist(E,F) | w**********r
发帖数: 128 | 2 Let x_n, y_n be a minimizing sequence of the distance with x_n in E and y_n
in F. Since E is compact, x_n is bounded, therefore y_n is bounded. Now use
the assumption that X is R^n, which means you can pass to a subsequence of x
_n ,y_n to get the limit. |
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