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 . Let be arbitrary but fixed.
If x is the greatest element of X, then .
If x is not the greatest element of X, then (*) and .
Then I claim that is the immediate successor of x, i.e. there doesn't exist a such that (**)
1. Does the proof depend on the choice of well-ordering?
2. How do I show (*)?
3. How do I show (**)?