Product of countable sets is countable

Statement

Let $A$ and $B$ be countable sets. Then the Cartesian product $A \times B$ is countable.

Remarks

Proof: take injections $f \colon A \to \mathbb{N}$ and $g \colon B \to \mathbb{N}$. Then $(a, b) \mapsto \pi(f(a), g(b))$ is an injection to $\mathbb{N}$, where $\pi$ is the Cantor pairing function. By induction, a finite product $A_1 \times \dots \times A_n$ of countable sets is countable. An infinite product need not be countable: the set of sequences of $0$ and $1$ is uncountable.

Explanations

L1 human

The pairs of natural numbers fill an infinite grid, and the grid can be listed diagonal by diagonal. So a product of two lists is again a list.