$\mathbf{Question:}$ $(H,+)$ is a subgroup of $(\mathbb{R},+)$ such that $H \cap [-1,1]$ is finite and contains elements other than $0$. Show that $(H,+)$ must be cyclic.
$\mathbf{Attempt:}$ Since $\{0\}\cup T= H \cap [-1,1]$ is finite [$0 \not\in T$, $\emptyset \subsetneq T$], so we can completely enumerate $T$. Let $T =\{a_1, a_2, ..., a_m\} $ be the complete 'list' of elements. Let $Q= \{a_i \in T: a_i>0 \}$. Let $a_t$ be the minimal element in $Q$.
Claim: $H=\langle a_t \rangle $.
Suppose, the claim is false. Then for some $h \in H$, $h \notin \langle a_t \rangle $. [we pick $h$ such that it is $>0$. If not, then we pick $-h$].
So, we can find a positive integer $p$ such that $p a_t< h <(p+1)a_t \implies 0<h-pa_t<a_t $ which contradicts the fact that $a_t$ is the minimal element in $Q$.
Is this Proof valid? Kindly verify.
Looks good to me. Maybe add an argument why $Q$ is not empty (I am sure that you understand why it is non-empty).