Claim: There exists a number such that
So, we look at the set . We want to consider . Claim that , and prove by contradiction.
There are two possibilites: or .
For : Let be large enough so that (<---Do not understand choice of n here). Then . Then
But then , and , contradicting the fact that is an upper bound for S.
This isn't the complete proof. You then have to use P.B.C. for the other case, . But, my problem is that I don't understand the choice of n here. Of course it makes sense looking at the proof, but how does one know how to choose n for any other problem, say, for ?