Subgroup Generated by Subgroup

Theorem
Let $G$ be a group.

Let $H \le G$ be a subgroup of $G$.

Then:
 * $H = \gen H$

where $\gen H$ denotes the subgroup generated by $H$.

Proof
By definition of generated subgroup, $\gen H$ is the smallest subgroup of $H$ containing $H$.

Hence the result.