Subgroup of Real Numbers is Discrete or Dense

Theorem
Let $G$ be a subgroup of the additive group of real numbers.

Then one of the following holds:
 * $G$ is dense in $\R$.
 * $G$ is discrete and there exists $a \in \R$ such that $G = a \Z$, that is, $G$ is cyclic.

Proof
If $G$ is trivial, then $G$ is discrete and cyclic.

Let $G$ be non-trivial.

Because $x \in G \iff -x \in G$, $G$ has a strictly positive element.

Thus $G \cap \R_{>0}$ is non-empty.

Let $a = \map \inf {G \cap \R_{>0} }$.

Case 1
Suppose $a = 0$.

We will show that $G$ is dense in $\R$.

Let $x \in \R$ and $\epsilon > 0$.

Because $a = 0$, there exists $g \in G$ such that $0 < g \le \epsilon$.

Then:
 * $y = g \cdot \floor {\dfrac x g} \in G$, where $\floor {\, \cdot \,}$ denotes the floor function.

We have:

Thus $G$ is dense in $\R$.

Case 2
Suppose $a > 0$.

We show that $G = a \Z$, from which it follows that $G$ is discrete.

Let $g \in G$.

Then:
 * $g - a \cdot \floor {\dfrac g a} \in G$

By Real Number minus Floor, we have:

Because $a = \map \inf {G \cap \R_{>0} }$, we have:
 * $g = a \cdot \floor {\dfrac g a}$

Thus:
 * $g \in a \Z$