Compact First-Countable Space is Sequentially Compact

Theorem
Let $T = \left({X, \tau}\right)$ be a compact first-countable topological space.

Then every infinite sequence in $X$ has a convergent subsequence; i.e., $T$ is sequentially compact.

Proof
Let $\left \langle {x_n} \right \rangle_{n \in \N}$ be an infinite sequence in $X$.

We use a Proof by Contradiction.

That is, suppose that $\left \langle {x_n} \right \rangle_{n \in \N}$ does not have a convergent subsequence.

By Accumulation Point of Infinite Sequence in First-Countable Space is Subsequential Limit, it follows that $\left \langle {x_n} \right \rangle_{n \in \N}$ does not have an accumulation point in $X$.

Thus, for each $x \in X$, we can select an open set $U_x$ such that $x \in U_x$ and $U_x$ only contains $x_n$ for finitely many $n \in \N$.

The set $\mathcal U = \left\{{U_x : x \in X}\right\}$ is clearly an open cover for $X$.

By the definition of a compact space, there exists a finite subcover $\{{U_{x_1}, U_{x_2}, \ldots, U_{x_m}}\}$ of $\mathcal U$ for $X$.

Then $U_{x_1} \cup U_{x_2} \cup \cdots \cup U_{x_m}$ contains all of $X$ since it is a cover.

However, each open set in this union only contains $x_n$ for a finite number of $n$, and this is a finite union of sets. So, the union only contains $x_n$ for finitely many $n$.

This implies that $x_n \in X$ for only finitely many $n \in \N$.

This is a contradiction, since $\left \langle {x_n} \right \rangle$ is a sequence in $X$, and so $x_n \in X$ for all $n \in \N$.