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

Let $\displaystyle X$ be any set and suppose that $\displaystyle \mathcal{M}$ is a collection of subsets of $\displaystyle X$ 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 $\displaystyle \mathcal{M}$ is a member of $\displaystyle \mathcal{M}$.

2. Any subset of $\displaystyle X$ that is not disjoint with every member of $\displaystyle \mathcal{M}$ is contained in $\displaystyle \mathcal{M}$.

I'm working on the first one, but I have found nothing that indicates that $\displaystyle \mathcal{M}$ is closed under finite intersection. Any help is greatly appreciated.