Prove that a subgroup is cyclic Let $ H $ be the subgroup of the group $ (\mathbb R,+) $ such that $ H \cap [-1,1] $ is a finite set containing a nonzero element. Show that $H$ is cyclic.

Hint: Write $H^+ = \{h \in H; h > 0\}$, assuming $H \neq 0$. And that if $m = \inf H^+ $, with $m \neq 0$ then $H = m \mathbb Z$.
By definition for every $\epsilon > 0$ there exists $h \in H^+$ such that $$m \leq h < m + \epsilon $$
Say $m \notin H$. Then you get a increasing sequence $(h_n)$ such that $$\lim h_n = m$$
as $H$ is a subgroup we have that $0 < h_n - h_{n-1} \in H^+$ thus $h_n - h_{n-1} > m$, which is a contradiction. Conclusion, $m \in H$ and it follows that $m\mathbb Z \subseteq H$. 
Conversely, let $h \in H^+$. Then there exists $q, r \in \mathbb Z$ such that $$h = qm + r$$
where $0 \leq r < m $. As $m= \inf H^+$ it must be that $r = 0$, because $r \in H$. Therefore $h = qm $ and then $H = m\mathbb Z$.
