Show AxB is countable

Question

Want to show if A, B are countable $\implies$ AxB is countable. I know I want to show that the elements of AxB can be enumerated (hence, AxB is bijective to $\mathbb{N}$), but how can I proceed from the table of elements of AxB below?

Answer 1

Consider $c_{n(n-1)/2+k}=(a_n,b_{n+1-k})$ for $n\in\mathbb N$ and $k=1,\dots,n$. This enumerates all elements of $A\times B$, so $A\times B$ is countable.