Let $\displaystyle H$ be a subgroup of $\displaystyle (\mathbb{Z},+)$ with more than one element, and let $\displaystyle m$ be the smallest positive integer in $\displaystyle H \cap \mathbb{N}$. Prove that $\displaystyle H=m\mathbb{Z}$. [Hint: Use the Division Algorithm]

Question: Does $\displaystyle m\mathbb{Z}$ look like this:

$\displaystyle m\mathbb{Z}=\{0, \pm m, \pm 2m, \pm3m, \cdot \cdot \cdot\}$?