TITLE: Dense uncountable proper subgroup of $(\mathbb{R},+)$
QUESTION [12 upvotes]: Probably someone had asked this question on StackExchange, but can one construct a dense uncountable proper subgroup of $(\mathbb{R},+)$?

REPLY [7 votes]: Here's one from Hardy's "A course of pure math":
$\displaystyle \{x \in \mathbb{R}: \lim_{n \to \infty} \sin(n! \pi x) = 0\}$.