# Union of an infinite set and a countable set

Let $A$ be an infinite set and $B$ be a countable set. I want to show that $|A|=|A\cup B|$.

I'm aware that the following relevant definitions:

• An infinite set is one which has a proper subset of equal cardinality.
• A countable set is one which is finite or countably infinite.
• We have $|A|=|A\cup B|$, for sets $A$ and $B$, precisely when there exists a bijection $A\to A\cup B$.

Furthermore, it seems like it will be easier to use the Cantor-Berstein theorem rather than explicitly constructing a bijection. However I'm not sure if using surjective maps is more effective than injective maps (vice versa) for solving this problem. Another important fact is that if $A$ is infinite and $|A|=|A'|$ for some set $A'$, then $A'$ is also infinite.