Use the "first form" of Principle of Math Induction to prove the Well Ordering Principle.

Question

Use the "first form" of Principle of Math Induction to prove the Well Ordering Principle. How can I do this? I tried doing it but I don't know where to start. Please I would I appreciate an explained answer rather than a hint. Thanks a bunch.

Answer 1

Suppose $S\subseteq \mathbb{N}$ is nonempty and has no minimum element. We know $1\not\in S$ since $1$ is a minimum element. Assume none of $1,2,..,k, k\in\mathbb{N}$ is an element of $S$. Since $S$ has no minimum element, $k+1\not\in S$. By induction, $S$ is empty. Thus, $\mathbb{N}$ is a well-order.