Countably Compact Metric Space is Compact/Proof 2

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

Let $M$ be countably compact.

Then $M$ is compact.

Proof
We have that a Countably Compact Metric Space is Sequentially Compact.

Then we have that a Sequentially Compact Metric Space is Separable.

For each $n$, a metric space which is countably compact can be covered by finitely many open $\epsilon$-balls : $N_{1/n} \left({x_i}\right)$.

So $\left\{{x_i}\right\}$ is a dense subset of $A$ which is countable.

So if a metric space is countably compact it is by definition second-countable.

The result follows from Second-Countable Space: Compactness Equivalent to Countable Compactness.