A Countable Union of Countable Sets is Countable

Let $S$ be a set, let $\mathbb{N}$ be the set of natural numbers, and let $(X_{m})_{m\in\mathbb{N}}$ be a family of subsets of $S$ such that $X_{m}$ is countable for every $m\in\mathbb{N}$.

Then the union $\bigcup_{m\in\mathbb{N}}X_{m}$ is countable.