Can someone explain this well ordering proof? Or explain where it's wrong since my dog jumped on my keyboard and messed up part of my notes from class (lol).

Well Ordering Principle (W.O.P.)Theorem:

Every non-empty subset of the ℕ (natural numbers) has a smallest element

Let S be a subset of ℕ, S ≠ ∅ (empty set)Proof:

Since S is bounded below, the infemum of S exists and is a real number.

Let ∝ = inf S = glb S.

If ∝ ɛ S then we are done.

Suppose ∝ is not ɛ S.

There must be an element of S smaller than ∝+1

Since ∝ is the greatest lower bound of S & x is not in S this element cannot be ∝ since ∝ is not in S

Ǝ x ɛ S ⋺ ∝<x<∝+1

NOTE: that x is not a lower bound for S since x>∝ and ∝ is the greatest lower bound of S

Therefore, Ǝ y ɛ S ⋺ ∝ <x<y<∝+1

Therefore, 0<x-y<1#

This is a contradiction since x and y ɛ ℕ

I really think it is messed up a bit somewhere but just started the class and did not get a good chance to look at it before it got messed up...

Any help is greatly greatly appreciated!!!