Exercise 5, § 6

Let $\mathfrak{F}$ and $\mathfrak{G}$ be two filters on a set $X$. Show that if $\mathfrak{F}$ and $\mathfrak{G}$ have a least upper bound in the set of all filters on $X$, then this least upper bound is the set of all $M \cap N$, where $M \in \mathfrak{F}$ and $N \in \mathfrak{G}$.