Let $G$ be a non-trivial subgroup of the group $(\mathbb{R}, +)$. Show that either $G$ is dense in $\mathbb{R}$ or that $G = \mathbb{Z} \cdot l$ where $l = \inf\{ x \in G \mid x > 0 \}$.
Let $G$ be a non-trivial subgroup of the group $(\mathbb{R}, +)$. Show that either $G$ is dense in $\mathbb{R}$ or that $G = \mathbb{Z} \cdot l$ where $l = \inf\{ x \in G \mid x > 0 \}$.