Claim: There exists a number such that

Proof:

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

So

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 ?