Prove that a subgroup is dense in $\mathbb{R}$

Question

Let $G$ be a subgroup of $(\mathbb{R}, +)$. Prove that $G$ is dense in $\mathbb{R}$ or any element in $G$ has the form $a\mathbb{Z}$, with $a \in \mathbb{R}$. (We denote that a group $G$ is dense in $\mathbb{R}$ if any element in $\mathbb{R}$ is the limit of a sequence of elements from $G$)

I haven't done anything notable yet.

Thank you!

Answer 1

Hint

Consider

$$\alpha := \inf \{x\in G\cap \mathbb R_+^*\}.$$

Then distinguish the two cases:

• $\alpha>0$ : prove then that $G=\alpha \mathbb Z$,

• $\alpha=0$ : prove then that $G$ is dense in $\mathbb R$.