Solution
= Solution
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 on a set>[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 $n$.