Proof That Countably Compact Subsets of a Metric Space are Also Sequentially Compact

Theorem

Any countably compact subsetof a metric spaceis also sequentially compact.

Proof

Letbe a metric space and letbe a sequence inIf the setis finite then one of the elements, sayappears infinitely often, hence the subsequenceconverges to

Suppose the setis infinite.is countably compact. The infinite subset ofhas an accumulation pointSinceis a metric space, we can choose a subsequencewhich converges toHenceis sequentially compact.