What does "prove the maximum principle" mean?
The question is as follows:
My half-baked idea:Use the well ordering principle to prove the maximum principle
I'm not sure how we can go further with this. What else can we prove with this information? What am I missing?Let:
and (all natural numbers)
There is a hint which states we should let a new set be the upperbounds of in (natural numbers)
We know , so by WOP it must have a minimum, which we shall call .
We then assume:
It's actually quite simple though. Prove that if you have a subset (of the natural numbers) which is non-empty and bounded, that the subset must have a maximum.
Then basically you assume it does not have a maximum and do a proof by contradiction that the earlier assumption was incorrect.
To continue from yesterday:
Which means must be a maximum for the set and we have the necessary contradiction.