Property of the additive group of reals

Question

Let $(\mathbb{R},+)$ be the standard additive group of reals. Then, if a subgroup of $(\mathbb{R},+)$, $H$ be such that $H\cap [-1,1]$ is finite, then is $H$ cyclic?

I am dumbstruck on this one? I need to show that all elements of $H$ are generated by a single element. I know that any subgroup of $(\mathbb{R},+)$ is either dense or of the form $m\mathbb{Z}$ for some $m>0$. Any hints. Thanks beforehand.

Answer 1

Hint: $H$ cannot be dense if $H\cap[-1,1]$ is finite.