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

If I have $G$ a closed subgroup of $\mathbb{R}$, then why is $G$ necessarily countable, except of course in the case where $G=\mathbb{R}$?

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    If the subgroup is dense and closed then must be everything. If it's not dense, it has a least positive element and the arguments in the linked thread show that that this element generates the subgroup.2011-12-11

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