TITLE: Subgroups of $(\mathbb R, +)$ are either dense or cyclic.
QUESTION [13 upvotes]: I was trying to prove that any subgroup of $(\mathbb R, +)$ is either dense in $\mathbb R$ or is a cyclic subgroup of $(\mathbb R, +)$. 
Thanks in advance for any help.

REPLY [6 votes]: Assume $G<\Bbb R$ is not  dense, say no element of $G$ is in the non-empty open interval $(a,a+\epsilon)$ with $\epsilon>0$.
Then show that for every $g\in G$, we have $G\cap (g-\epsilon,g+\epsilon) =\{g\}$.
Then show that $G\cap(0,\infty)$ is either empty (in which case $G=\{0\}$ is cyclic) or has a minimal element $a$ (in which case $G=\langle a\rangle$)