Hi, I need this problem to study off of. It is not for a grade. Please provide a complete prove if you can. Thanks,

Suppose $\displaystyle K_1 \supset K_2 \supset K_3 \supset $... is a descending sequence of nonempty sonnected subsets of a metric space M. Prove: if M is compact, the intersection $\displaystyle \bigcap K_n $ is a connected subset of M.

Also give a counterexample to show this is false without compactness.

Thanks.