Subgroup of finite group is finite

This article gives the statement, and possibly proof, of a basic fact in group theory.
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Statement

Suppose $G$ is a finite group and $H$ is a subgroup of $G$. Then, $H$ is also a finite group.

Proof

Suppose $G$ is a finite group and $H$ is a subgroup of $G$. The underlying set of $G$ is finite, and as $H$ is a subgroup of $G$, the underlying set of $H$ must also be finite. Hence $H$ is finite.