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Suppose K is a nonempty compact subset of a metric space X and x∈X. Show, there...

Suppose K is a nonempty compact subset of a metric space X and x∈X.

Show, there is a nearest point p∈K to x; that is, there is a point p∈K such that, for all other q∈K, d(p,x)≤d(q,x).

[Suggestion: As a start, let S={d(x,y):y∈K} and show there is a sequence (qn) from K such that the numerical sequence (d(x,qn)) converges to inf(S).] Let X=R^2 and T={(x,y):x^2+y^2=1}.

Show, there is a point z∈X and distinct points a,b∈T that are nearest points to z. Let X=(0,∞)∪{−1} (as a subspace of R). Show there is no nearest point to −1 from the closed (in X) set K=(0,1].

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