Subgroups of cyclic groups are cyclic

Statement

Let $G$ be a cyclic group and let $H$ be a subgroup of $G$. Then $H$ is cyclic.

Remarks

Let $g$ generate $G$. If $H \neq \{e\}$, let $m$ be the smallest positive integer with $g^m \in H$; then division with remainder shows that $H = \langle g^m \rangle$. For example, each subgroup of $(\mathbb{Z}, +)$ has the form $m\mathbb{Z}$, generated by $m$.

Explanations

L1 human

A subgroup of a cyclic group is generated by the smallest positive power of the generator that it contains. Any other power in the subgroup is a multiple of this one, or the subgroup would contain a smaller positive power. Thus the subgroup is again cyclic.