Given a non empty subset $\displaystyle S$ of $\displaystyle \mathbb{N}$
Show that either there is a unique $\displaystyle n\epsilon \mathbb{N}$ for which there is a unique increasing bijective function $\displaystyle u:n \to S$, or there is a unique increasing bijection $\displaystyle u:\mathbb{N} \to S$. Moreoevr $\displaystyle u$ is unique for this property.