Sequentially Compact Space is Countably Compact

Theorem
A sequentially compact topological space is also countably compact.

Proof
Let $T = \struct {S, \tau}$ be a sequentially compact topological space.

By definition of countably compact space, it suffices to show that every infinite sequence in $S$ has an accumulation point in $S$.

By the definition of a sequentially compact space, every infinite sequence in $S$ has a subsequence which converges in $S$.

The result follows from Limit of Sequence is Accumulation Point.