# Property of the additive group of reals

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.

• If you already know that a subgroup is either dense or cyclic, prove that a dense subgroup must intersect $[-1,1]$ in infinitely many points. – egreg Jan 28 '18 at 11:37
• @egreg thanks, quite a trivial issue! so foolish of me! – vidyarthi Jan 28 '18 at 11:41

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