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).
Theorem: Well Ordering Principle (W.O.P.)
Every non-empty subset of the ℕ (natural numbers) has a smallest element
Proof: Let S be a subset of ℕ, S ≠ ∅ (empty set)
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!!!