Proof that the product of countable sets is countable

Proof

1. Let $A$ and $B$ be countable sets. By definition, there are injective functions $f \colon A \to \N$ and $g \colon B \to \N$.
2. Define $h \colon A \times B \to \N \times \N$ on the Cartesian product by $h(a, b) = (f(a), g(b))$.
3. The map $h$ is injective. If $h(a, b) = h(a', b')$, then $f(a) = f(a')$ and $g(b) = g(b')$. So $a = a'$ and $b = b'$, since $f$ and $g$ are injective.
4. Let $\pi \colon \N \times \N \to \N$ be the Cantor pairing function, $\pi(x, y) = (x + y)(x + y + 1)/2 + y$. It is a bijection, so it is injective.
5. The composite $\pi \circ h \colon A \times B \to \N$ is injective, as a composition of injections.
6. Hence $A \times B$ is countable.