I have this kind of problem:
Let be a non-empty set, where is valid
with every finite .
Prove that there is the maximal element in the set .
My solution (or the idea at least):
First, let's choose arbitrary .
If is the chain, then holds or .
Now I'm facing the problem. I know that in set , there exists an element , which has every finite , where . (It occur me, that maybe . Is that possible?)
How can I prove, that there exists this element ?
After I have solved the problem above, I can proceed with my solution:
Because , then every ,where , also holds that every . That means, that with every finite . So .
And according to Zorn's lemma, there exists the maximal element in the set .