Subgroups generate themselves

Let $G$ be a group. Prove that if $H \leq G$ is a subgroup then $\langle H \rangle = H$.

That $H \subseteq \langle H \rangle$ is clear. Now suppose $x \in \langle H \rangle$. We can write $x$ as a finite product $h_1 h_2 \cdots h_n$ of elements of $H$; since $H$ is a subgroup, then, $x \in H$.