Set of Natural Numbers is either Finite or Denumerable

Theorem
Let $S$ be a subset of the natural numbers $\N$.

Then $S$ is either finite or denumerable.

Proof
Let $\omega$ denote the set of natural numbers as defined by the von Neumann construction.

By the Well-Ordering Principle, $\omega$ is well-ordered by the $\le$ relation.

Thus from the Well-Ordering Principle, $S$ has a smallest element.

Let this smallest element of $S$ be denoted $s_0$.

Also from the Well-Ordering Principle, every subset of $S$ also has a smallest element.