Hello, I am recently trying to learn the whole concept of Well-Ordering, and my teacher isn't one to show us how to do things, instead for us to figure it out.
I believe I get the concept, I just want to make sure I have my proof skeleton down and would welcome any input. Thanks.
Just for illustrative purposes, let P(x) = x^2 >= 2x.
Prove P(x) for all positive x in Z.
Let T be a non-empty set of positive integers such that P(x) is not true. By the Well-Ordering Principle T contains a smallest element k.
(This is my first trouble, I don't really understand how T can have a smallest element k, since P(x) holds for k. But I KNOW you are supposed to somehow show that k-1 also holds, and this contradicts that k is the smallest element. But it seems that you just have to show that T is empty, that is k isn't even in T, so T is empty, and therefore all positive integers hold for P(x))
plug in k in P(k) and show that it is true. Therefore , T is empty. Contradiction and P(x) must hold for all positive integers.
Thanks for any input, this is reeaaaaally killing me. After having mathematical induction drilled into me, I understand that, and Well-Ordering doesn't seem to make any sense other than that "duh" kind of way. I don't really know how to implement it in proofs.