Compactness and Sequential Compactness are Equivalent in Metric Spaces

Theorem

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

Then $\struct {X, d}$ is compact if and only if it is sequentially compact.

Proof

Necessary Condition

This is precisely Compact Subspace of Metric Space is Sequentially Compact in Itself.

$\Box$

Sufficient Condition

This is precisely Sequentially Compact Metric Space is Compact.

$\blacksquare$