Sequentially Compact Metric Subspace is Sequentially Compact in Itself iff Closed

Theorem
Let $$M$$ be a metric space.

Let $$C \subseteq M$$ be a subspace of $$M$$ which is sequentially compact in $$M$$.

Then $$C$$ is sequentially compact in itself iff $$C$$ is closed in $$M$$.

Proof
Follows directly from Closure of Subset of Metric Space by Convergent Sequence.