The study of symmetry: groups, subgroups, homomorphisms, and group actions.

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Let $H$ be a subgroup of the group $(R, +)$ such that $H$ $∩$ [-1,1] is a finite set containing a non zero element. Show that $H$ is cyclic.

Observations: Since $H$ is a subgroup of $(R, +)$ so $0 \in H.$ If $1 \in H,$ then all positive integers belong to $H.$ But $H$ is closed wrt addition, so the negative integers must belong to $H$ ...