Why is any proper closed subgroup of $\mathbb{R}$ necessarily countable?

Question

Possible Duplicate:
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 $G=\mathbb{R}$?