Proof That a Countably Compact Metric Space is Separable

Theorem

Any countably compact metric spaceis separable.

Proof

Letbe any positive number. There is a maximal subsetsuch that for

Supposeis infinite for somethenhas an accumulation point

contains infinitely many points ofAll points inare a distance less thanfromsocontains infinitely many points ofThis is a contradiction andis finite for each

Then for anyelse contradicting the maximality of

For each positive integer n definethenis a countable dense subset ofandis separable.