Principal Ultrafilter is All Sets Containing Cluster Point

Theorem
Let $S$ be a set.

Let $\mathcal P \left({S}\right)$ denote the power set of $S$.

Let $\mathcal F \subset \mathcal P \left({S}\right)$ be a principal ultrafilter on $S$.

Let its cluster point be $x$.

Then $\mathcal F$ is the set of all subsets of $S$ which contain $x$.