# Characterizing Dense Subgroups of the Reals

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Subgroup of $\mathbb{R}$ either dense or has a least positive element?

Let $(\mathbb{R},+)$ be the team of Real Numbers under enhancement. Allow $H$ be a correct subgroup of $\mathbb{R}$. Confirm that either $H$ is thick in $\mathbb{R}$ or there is an $a \in \mathbb{R}$ such that $H=\{ na : n=0, \pm{1},\pm{2},\dots\}$.

I am unable to continue.

If there is a tiniest favorable component, after that we are done, given that any kind of favorable component has to be an integer multiple of it, or otherwise we can make use of a euclidean - type - algorithm to get a favorable component with smaller sized value. (I.e., intend $a$ is the tiniest favorable component, and also $b$ a favorable component which is not an integer multiple of $a$ - - - maintain deducting duplicates of $a$ till you get something that is purely in between $0 $ and also $a$.)

So think there is a series $a_n$ had in the team that contains favorable numbers often tending to absolutely no. After that the team has each ${\mathbb{Z} a_n}$. This suggests that for each and every $n$, any kind of number in $\mathbb{R}$ is within $|a_n|$ of a component of the team. Given that the $|a_n|$ can be tiny, we locate that the team is thick.

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