Hello noles2188In fact,issuch a set (*see the Note below), so we must assume that .

So, let us assume that , and . (In other words, let's assume the opposite of what we want to prove.)

Then let be the propositional function:

Then by hypothesis

Now is true for some

So, by induction is true

is true . But . Contradiction. Therefore the original assumption is false.

Grandad

*Note: the proposition is true whenever is false. So if , the proposition is true, because is always false. It seems a bit crazy at first sight, but it is OK!