principle of finite induction proven from the well-ordering principle for natural numbers

We give a proof for the “strong” formulation.

Let $S$ be a set of natural numbers such that $n$ belongs to $S$ whenever all numbers less than $n$ belong to $S$ (i.e., assume , where the quantifiers range over all natural numbers). For indirect proof, suppose that $S$ is not the set of natural numbers $\mathbb{N}$. That is, the complement $\mathbb{N}\setminus S$ is nonempty. The well-ordering principle for natural numbers says that $\mathbb{N}\setminus S$ has a smallest element; call it $a$. By assumption, the statement holds. Equivalently, the contrapositive statement holds. This gives a contradition since the element $a$ is an element of $\mathbb{N}\setminus S$ and is, moreover, the smallest element of $\mathbb{N}\setminus S$.