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 . 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 (**)?