Well ordering principle and the maximum principle

Hello chaps (Hi)

The question is as follows:

Quote:

Use the well ordering principle to prove the maximum principle

My half-baked idea:

Quote:

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:

I'm not sure how we can go further with this. What else can we prove with this information? What am I missing?