Immediate successors in Well-ordered sets

A little help with this question?

Let http://alt2.artofproblemsolving.com/...6beadf144b.gif be a well-ordered set. Show that for each http://alt2.artofproblemsolving.com/...2aaea33e95.gif, either http://alt1.artofproblemsolving.com/...03785c2072.gif is the greatest element of http://alt2.artofproblemsolving.com/...6beadf144b.gif or http://alt1.artofproblemsolving.com/...03785c2072.gif has an immediate successor (that is, there exists http://alt2.artofproblemsolving.com/...85b7ebf3e8.gif such that there does not exist http://alt1.artofproblemsolving.com/...abf4293bdb.gif with http://alt1.artofproblemsolving.com/...3194b92936.gif )

Now, here's how I started:

Let X be well-ordered by $\displaystyle \leq $. Let $\displaystyle x \in X $ be arbitrary but fixed.

If x is the greatest element of X, then $\displaystyle \forall y \in X, y \leq x $.

If x is not the greatest element of X, then $\displaystyle x^+ \in X $ (*) and $\displaystyle x \leq x^+ $.

Then I claim that $\displaystyle x^+ $ is the immediate successor of x, i.e. there doesn't exist a $\displaystyle y \in X $ such that $\displaystyle x<y<x^+ $ (**)

Questions:

1. Does the proof depend on the choice of well-ordering?

2. How do I show (*)?

3. How do I show (**)?