TITLE: Characterizing Dense Subgroups of the Reals
QUESTION [13 upvotes]: Possible Duplicate:
Subgroup of $\mathbb{R}$ either dense or has a least positive element? 

Let $(\mathbb{R},+)$ be the group of Real Numbers under addition. Let $H$ be a proper subgroup of $\mathbb{R}$. Prove that either $H$ is dense in $\mathbb{R}$ or there is an $a \in \mathbb{R}$ such that $H=\{ na : n=0, \pm{1},\pm{2},\dots\}$.
I am not able to proceed.

REPLY [18 votes]: If there is a smallest positive element, then we are done, since any positive element must be an integer multiple of it, or otherwise we could use a euclidean-type-algorithm to get a positive element with smaller value. (I.e., suppose $a$ is the smallest positive element, and $b$ a   positive element which is not an integer multiple of $a$---keep subtracting copies of $a$ until you get something that is strictly between $0 $ and $a$.)
So assume there is a sequence $a_n$ contained in the group that consists of positive numbers tending to zero. Then the group contains each ${\mathbb{Z} a_n}$. This means that for each $n$, any number in $\mathbb{R}$ is within $|a_n|$ of an element of the group. Since the $|a_n|$ can be small, we find that the group is dense.