Compact Metric Space is Separable

Theorem

Let $\struct {X, d}$ be a compact metric space.

Then $\struct {X, d}$ is separable.

Proof

From Compactness and Sequential Compactness are Equivalent in Metric Spaces, $\struct {X, d}$ is sequentially compact.

From Sequentially Compact Metric Space is Separable, $\struct {X, d}$ is separable.

$\blacksquare$