Countably Compact Metric Space is Compact/Proof 1

Theorem
Let $M = \left({A, d}\right)$ be a metric space.

Then $M$ is countably compact iff $M$ is compact.

Proof
We have from Compact Space is Countably Compact that compactness implies countable compactness in all topological spaces regardless of whether they are metric spaces or not.

So all we need to do is demonstrate that if $M$ is countably compact then it is compact.

We have that a countably compact metric space is sequentially compact.

Then we have that a sequentially compact metric space is totally bounded.

A totally bounded metric space is second-countable.

The result follows from the fact that in a second-countable space, compactness is equivalent to countable compactness.