I'm working on a problem, but I just can't figure it out.

Let be any set and suppose that is a collection of subsets of that is maximal with respect to the finite intersection property. Prove the following statements are true.

1. The intersection of any finite nonempty subcollection of is a member of .

2. Any subset of that is not disjoint with every member of is contained in .

I'm working on the first one, but I have found nothing that indicates that is closed under finite intersection. Any help is greatly appreciated.