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a convergent sequence

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Let (X,d) be a metric space with x in X and A as a nonempty subset of X. The distance between x and A is defined as:
dist(x,A) = inf{d(x, a) : a in A}

i > 0

A_i = {x in X : dist(x,A) <= i}

a)
Show that A_i is closed

b)
C is a collection of all compact subsets of X.
C is nonempty
p : C Ã? C -> [0,infinity)
p(A,B) = inf{i > 0 : B subset of A_i and A subset of B_i}

Show that p is a metric on C when C is nonempty

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Solution Summary

This solution explains how to provide a series of proofs for a convergent sequence.

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