A little help with this question?

Let

be a well-ordered set. Show that for each

, either

is the greatest element of

or

has an immediate successor (that is, there exists

such that there does not exist

with

)

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,

This isn't usually true
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 (**)?