Exercise 12, § 3

Let X be a set, $\mathscr{U}$ a subset of $\mathscr{P}(X)$, and $m$ the characteristic function of $\mathscr{U}$. For $m$ to be a gauge, it is necessary and sufficient that $\mathscr{U}$ be an ultrafilter and that the intersection of every countable subset of $\mathscr{U}$ belong to $\mathscr{U}$. Under these conditions, $\mathscr{U}$ is said to be an Ulam ultrafilter on X. Suppose that this is the case, and that $(H_i)_{i \in I}$ is a family of elements of $\mathscr{U}$ such that $\mathrm{Card}(I)$ is an Ulam cardinal; show that $\bigcap_{i \in I} H_i$ belongs to $\mathscr{U}$.