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.

$\blacksquare$

Also see

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

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Exercise $17.12$
• 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 7$: Countable and Uncountable Sets: Corollary $7.3$