# Compact implies Sequentially Compact

Homework Statement [/b]
I need help proving that if X is a metric space and E a subset of X is compact, then E is sequentially compact.

I know I need to consider a sequence x_n in E, and I want to say that there is a point a in E and a radius r > 0 so that Br(a) [the ball of radius r with center a] contains x_k for infinitely many k's. If I show this, then I think I can conclude that any subsequence of x_n converges to a. Can someone please help?

Hurkyl
Staff Emeritus
Gold Member
So... your conjecture is that every sequence in a compact metric space is actally convergent?

No, that is not one of the assumptions...my second paragraph in the problem description is a hint in the back of the textbook as to how to go about the problem.

Office_Shredder
Staff Emeritus
Gold Member
If any subsequence of xn converges to a, then xn converges to a too. Rather, what you can conclude is a single subsequence of xn converges to a (which is what sequential compactness is about)

If any subsequence of xn converges to a, then xn converges to a too. Rather, what you can conclude is a single subsequence of xn converges to a (which is what sequential compactness is about)

This is not true- consider a_n=(-1)^n=1,-1,1,-1,1,-1,...

Then a_{2n}=1 is a subsequence that converges to 1, but a_n does not converge at all- nonetheless to 1.

HallsofIvy