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:

$\exists n \in \N: \card S = n$

where $\card S$ denotes the cardinality of $S$.

From Cardinality of Power Set of Finite Set:

$\card {\powerset S} = 2^n$

where $\powerset S$ denotes the power set of $S$.

As $n \in \N$ it follows that $2^n \in \N$ and so $\powerset S$ is also by definition a finite set.

$\blacksquare$