Accumulation Point of Infinite Sequence in First-Countable Space is Subsequential Limit

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

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

Let $x$ be an accumulation point of $\left\langle{x_n}\right\rangle$.

Then $x$ is a subsequential limit of $\left\langle{x_n}\right\rangle$.