$A, B$ --- enumerable sets, is $A \times B$ enumerable?
There are many ways you could solve this, so here's a hint. $A$ and $B$ are enumerable, which means that injections

$$ f : A \rightarrow \mathbb{N} \quad \text{and} \quad g : B \rightarrow \mathbb{N}$$

exist. Now, given a tuple $t \in A \times B$, how can you map it to a unique element in $\mathbb{N} \times \mathbb{N}$ using $f$ and $g$?


   Since $f$ and $g$ are injections, their product must also be an injection (prove this). So the function $h : A \times B \rightarrow \mathbb{N} \times \mathbb{N}$, defined as $(a, b) \mapsto (f(a), g(b))$. Is an bijection on a subset of $\mathbb{N} \times \mathbb{N}$ (because it is an injection on $\mathbb{N} \times \mathbb{N}$).


A subset of an enumerable set is always countable, so this concludes the proof.