I have this kind of problem:

Let

be a non-empty set, where is valid

with every finite

.

So A is a set whose elements are sets...?
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,

You mean "__a__ 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

.