Subset of Countable Set is Countable

Theorem
A subset of a countable set is countable.

Proof
Let $S$ be a countable set.

Let $T \subseteq S$.

By definition, there exists an injection $f: S \to \N$.

Let $i: T \to S$ be the inclusion mapping.

We have that $i$ is an injection.

Because the composite of injections is an injection, it follows that $f \circ i: T \to \N$ is an injection.

Hence, $T$ is countable.

Also see

 * Subset of Countably Infinite Set is Countable, a special case of this theorem