Increasing enumeration of an infinite subset of natural numbers ($f(0)=\min A$)

For an infinite subset $A\subseteq \mathbb{N}$, repeatedly selecting the least element not yet used gives a bijection $f:\mathbb{N}\to A$. The well-order ensures each minimum exists, infinitude ensures the remaining set is nonempty, and strict increase gives injectivity. If $a\in A$ were omitted forever, all selected values would be at most $a$, impossible inside the finite set $\{0,\dots ,a\}$.