Problem 1. (10 points) In lecture we proved that the Principle of Induction follows from the Well-ordering principle. In this problem we will prove the converse; namely, that PI WOP. Nothing could be simpler, thought Lem E. Hackett. He had proved tougher theorems previously. Here's the hypothesis that Lem came up with: P(k): Every set containing k 2 1 elements has a least element. Lem quickly established P(1), the base case. He then established P(k) P(k + 1) as follows: If you remove one element from any set with k + 1 elements, then you are left with a set with k elements. By the inductive hypothesis, this has a least element. We can compare this with the element that was removed; the smaller one is the least in the original set of k +1 elements. Done! Or so, Lem figured. Alas, his professor told him that although there was no flaw in his argument, the problem was that Lem had not established the well-ordering principle. (a) why did Lem's proof not establish PI WOP? (b) Thinking it over, Lem figured out a different approach. If the well-ordering principle is false, then there must be a non-empty set A that does not contain a least element. So Lem changed his hypothesis to: P(k): k EA. Help complete the proof for Lem