College level proof before real analysis. Please give formal proof.
Please explain each step of your solution.
Prove that the Well-Ordering Principle is equivalent to the Principle of Mathematical Induction
Well-ordering principle has two phases.
Phase 1: Each nonempty subset of natrual number N contains a smallest element.
Phase 2: To prove that every natural number belongs to a specified set S, assume not, then there is a nonempty subset H
of N that does not belong to S. For this subset H, from phase 1, we have the smallest element k. To make a contradiction,
we need to show that either k belongs to S or ...
This is a proof regarding well-ordering principle and principle of mathematical induction.