Cyclicity of some special subgroup of $(\mathbb R,+)$

Question

If $H$ is a subgroup of $(\mathbb R,+)$ such that $H\cap[-1,1]$ is finite and contains a positive element then is it true that $H$ is cyclic ?

Answer 1

Given that $H\cap [-1,1]$ is finite. Then, the set contains a least positive element. Let's say $a$ is this element.

Claim: $H=\langle a \rangle.$

Suppose, $\exists b\in H$ such that $b$ not in $\langle a\rangle$.

Apply division algorithm and try to get to a contradiction.

Answer 2

1. Suppose that $H$ is not cyclic and it has at least $2$ $\mathbb{Q}$-independent generators $g_1, g_2$.

2. Without loss of generality one of them $g_1 = 1$ (otherwise we can assume subgroup $g_1^{-1}H$ and it isomorphic to $H$). From this we can say that $g_2$ is irrational.

3. Finally, $\forall x \in \mathbb{R} \backslash \mathbb{Q}$ and $ \forall m,n \in \mathbb{N}$ $nx \neq mx$ (mod $1$) (otherwise $x$ is rational). So all residues $nx$ modulo $1$ are different. Contradiction with fact that $H \cap [-1, 1]$ is finite.