Powers of Element form Subgroup

Theorem
Let $\struct {G, \circ}$ be a group.

Then:
 * $\forall a \in G: H = \set {a^n: n \in \Z} \le G$

That is, the subset of $G$ comprising all elements possible as powers of $a \in G$ is a subgroup of $G$.

Proof
Clearly $a \in H$, so $H \ne \O$.

Let $x, y \in H$.

Thus by the One-Step Subgroup Test:
 * $H \le G$