Compact Metric Space is Separable
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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.
$\blacksquare$