Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 16H / c / i

The family ${\mathcal{U}}_n=\{A\subseteq \mathbb{N}:n\in A\}$ is nonempty, excludes the empty set, is closed under finite intersections, and is upward closed. It is therefore a filter. For every $A\subseteq \mathbb{N}$, either $n\in A$ or $n\in A^c$, so either $A\in {\mathcal{U}}_n$ or $A^c\in {\mathcal{U}}_n$. Hence ${\mathcal{U}}_n$ is the principal ultrafilter at $n$.