Say you have a set $A_i$ for $i$ in the natural numbers $\mathbb{N}$, and that is a countable set. Then for all natural numbers $n$, the union of those sets is countable.
I must prove this by induction, and I do realize that to do that, I must show that there is an injection( or surjection ) between the first two sets. Everything should follow accordingly afterwards, but I had the idea to utilize Cantor's Diagonal Argument in the inductive proof, but I am not sure how to go about defining injections for the argument in a way that shows that the sets are countable.