The rationals are countable

Statement

The set $\mathbb{Q}$ of rational numbers is countably infinite.

Remarks

Proof: each rational number has a unique form $p/q$ with $p \in \mathbb{Z}$, $q \ge 1$ and $\gcd(p, q) = 1$. The map $p/q \mapsto (p, q)$ is an injection into $\mathbb{Z} \times \mathbb{N}$, which is countable. And $\mathbb{Q}$ is infinite. The rationals are dense in $\mathbb{R}$, but they are countable, and $\mathbb{R}$ is not countable.

Explanations

L1 human

Write the fractions $p/q$ in a grid, with $p$ along one side and $q$ along the other. List the grid diagonal by diagonal, and skip each fraction that is not in lowest terms. Each rational number appears exactly one time.