Infinite pigeonhole principle

Theorem (Infinite Pigeonhol Principle) Suppose the set $P$ is infinite, $H$ is finite, and $f:P\mapsto H$. Then there is an element $h\in H$ such that the preimage set $P_h:=\{p\in P|f(p)=h\}$ is infinite.

Proof By contradiction, suppose that every such preimage set is finite. The set $P$ can be written as:

$$P=\bigcup _{h\in H}{P}_n$$

but a finite union of finite set is finite. $\square$