1. The problem statement, all variables and given/known data Prove Mathematical Induction using the Well-Ordering Principle. 2. Relevant equations None. 3. The attempt at a solution Every solution I can think of, and every solution I've seen, at some point along the proof, seems to me to employ the very reasoning that is used when we prove problems by mathematical induction, the only difference in this proof is we add in the method of proof by contradiction. So, in other words, it seems to me the only way to accomplish a solution is by employing circular reasoning. That doesn't strike me as a very satisfying way to solve this problem. So, to be clear, I'm not asking for a solution to be handed to me, I just want to know if there is actually a way to accomplish this proof that doesn't beg the question, i.e. that does not employ mathematical inductive reasoning?