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.

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