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:

$\displaystyle S \neq \emptyset$ and $\displaystyle S \subset N$ (all natural numbers)

There is a hint which states we should let a new set $\displaystyle (B)$ be the upperbounds of $\displaystyle S$ in $\displaystyle N$ (natural numbers)

We know $\displaystyle B \neq \emptyset$, so by WOP it must have a minimum, which we shall call $\displaystyle m$.

We then assume:

$\displaystyle x < m \forall x \in S$

$\displaystyle x + 1 \leq m \Rightarrow x + 1 \leq B$

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