I need help understanding a certain part of the following proof.

Theorem: (Monotone Class Theorem)

Let be a class of subsets of , closed under finite intersections and containing . Let be the smallest class containing which is closed under increasing limits and by difference. Then .

Proof:

The intersection of classes of sets closed under increasing limits and differences is a class of that type. So by taking the intersection of all such classes, there always exists a smallest class containing which is closed under increasing limits and by differences. For each set , denote

to be the collection of sets such that and .

Clearly is closed under increasing limits and by difference.

Let ; for each one has

and , so .

HERE THE PROBLEM STARTS

Now let . For each we have , and because of the preceeding

, hence , whence , hence .

The rest of the proof is just concluding that is a -algebra.

Back to the problem part. Why must ?

It's easy to see when , because is closed under finite intersections.