Hello chaps

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: