Power Set of Finite Set is Finite

Theorem
Let $S$ be a finite set.

Then the power set of $S$ is likewise finite.

Proof
Let $S$ be a finite set.

Then by definition:
 * $\exists n \in \N: \left|{S}\right| = n$

where $\left|{S}\right|$ denotes the cardinality of $S$.

From Cardinality of Power Set of Finite Set:
 * $\left|{\mathcal P \left({S}\right)}\right| = 2^n$

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

As $n \in \N$ it follows that $2^n \in \N$ and so $\mathcal P \left({S}\right)$ is also by definition a finite set.