Past exam of the mathematics course of the University of Cambridge / 2015 / ia / Paper 4 / 8E / b / i / Solution

Choose injections $a:X\to \mathbb{N}$ and $b:Y\to \mathbb{N}$. The map $(x,y)\mapsto (a(x),b(y))$ is an injection into the countable set $\mathbb{N}\times \mathbb{N}$. Composing with the Cantor pairing function gives an injection into $\mathbb{N}$. Thus the Cartesian product of two countable sets is countable, including the case of an empty factor.