Finite Space is Sequentially Compact

Theorem

Let $T = \struct {S, \tau}$ be a topological space where $S$ is a finite set.

Then $T$ is sequentially compact.

Proof

We have:

Finite Topological Space is Compact

Compact Space is Countably Compact

Finite Space is Second-Countable

Second-Countable Space is First-Countable

First-Countable Space is Sequentially Compact iff Countably Compact.

Hence the result.

$\blacksquare$