Denumerable and infinite sets > If $A$ is an infinite set and $B$ is denumerable, $A$ is equipotent with the union $A\cup B$. How to prove this?

Let $B'=B\setminus A$. If $B'$ is finite, then biject $B$ with $A\cap B$, and use that bijection to biject $A\cup B = A\amalg B'$ with $A$ (where $\amalg$ denotes a disjoint union).

If $B'$ is infinite, let $D$ be a denumerable subset of $A$. Then biject $D\amalg B'$ with $D$, and use that bijection to get a bijection of $A\cup B = A\amalg B'$ with $A$.

(This requires you to prove that the union of two disjoint denumerable sets is denumerable; I trust you know how to do that)