Countably Compact Metric Space is Compact/Proof 2

Proof
We have that a Metric Space is Countably Compact iff 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 $\paren {1/n}$-balls: $\map {B_{1/n} } {x_i}$.

So $\set {x_i}$ 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 is Compact iff Countably Compact.