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.
Are you talking about a finite union of countable sets? If so, for the induction hypothesis write $$ \cup_{i=1}^n A_i=(\cup_{i=1}^{n-1}A_i)\cup A_n. $$ Now both sets on the right hand side are countable, so there is a surjection from $\mathbb{N}$ to each of them. Can you turn this into a surjection from $\mathbb{N}$ to their union, possibly by splitting $\mathbb{N}$ into $\mathbb{N}=\{2k:k\in \mathbb{N}\}\cup \{2k+1:k\in \mathbb{N}\}$?