Here's the problem
Show that S, subset R, is compact iff every infinite subset of S contains at least one accumulation point.
I've proven the => part:
Suppose that S was compact. THen, let T be an infinite subset of S. Then, since every subset of a compact set is compact, T is compact. Hence, T is bounded. Thus by the Bolzano-Weierstrass theorem, T has an accumulation point.
But, I'm having a good deal of difficulty proving the converse.