Hello jzelltWe can do this with a combination of proof by induction and reductio ad absurdum.

Assumption:Assume that has no maximum. Then .

We now show by induction that .

So, define the propositional function as . Then

So .

is true. So by induction is true .

In particular if , where is an upper bound of , . But . Contradiction.

So our original assumption is false, and A therefore has a maximum.

Grandad