Let AcX, A is not the empty set.

Prove if every b in X there exists an a in A such that d(b,A)=d(b,a) then A is a closed subset of X.

So for A to be a closed subset, that means that it has to include all of its limit points. So should I assume that b is also in A?

I was thinking that I could say that A has a sequence {an} that converges to a, but how does that relate to b? with d(b,A)=d(b,a)? I am not sure how to put the two together.