Addition of infinite cardinal numbers

Question

Recently while trying to prove the Schröder–Bernstein theorem, I stumbled upon a fact that, if $m$ and $n$ are two cardinal numbers such that atleast one of them is infinite, then $m+n= \max\{m,n\}$. Now I can prove this fact when say, $m$ is any infinite cardinal and $n \leq \aleph_0$. But I can't go beyond that. So I searched a bit for a more general proof but couldn't find any. So any information related to this will be highly appreciated. Thank you.