Every nontrivial discrete subgroup of reals is infinite cyclic

Statement
Let $$H$$ be a nontrivial discrete subgroup of $$\R$$, the additive group of real numbers. Then, $$H$$ is an infinite cyclic group; it is described as $$m\mathbb{Z}$$ where $$m$$ is the smallest positive real number in $$H$$.

Related facts

 * Every nontrivial subgroup of the group of integers is cyclic on its smallest element
 * Every discrete subgroup of Euclidean space is free Abelian on a linearly independent set

Facts used

 * Discrete subgroup implies closed: Any discrete subgroup of a $$T_0$$-topological group is closed.

Proof
Given: A nontrivial discrete subgroup $$H$$ of $$\R$$

To prove: $$H$$ is an infinite cyclic group, generated by $$m$$, where $$m$$ is the smallest positive real number in $$H$$

Proof: Since $$H$$ is nontrivial, it contains a nonzero element. Further, since $$H$$ is a subgroup, it contains a nonzero positive element (if the element we start with is negative, we can take its additive inverse).

Consider the set of all positive elements of $$H$$. This is a nonempty set, so it has an infimum. Moreover, since $$H$$ is discrete, there exists a neighbourhood of $$0$$ not intersecting $$H$$, hence the infimum is strictly greater than $$0$$. Since $$H$$ is a closed subgroup of $$\R$$, this infimum must itself be in $$H$$, and is hence the smallest element of $$H$$. Call this element $$m$$.

Now suppose $$a$$ is any element of $$H$$. We can write:

$$a = mq + r$$

where $$q$$ is an integer and $$0 \le r < m$$. Then, $$r = a - mq \in H$$, since $$a,m \in H$$. Since $$m$$ is the smallest positive real in $$H$$, $$r = 0$$. Thus, every element of $$H$$ is a multiple of $$m$$, completing the proof.