Open subgroups of $\mathbb{R}$ Let $G$ be a nonempty open subset of $\mathbb{R}$ (with usual topology on $\mathbb{R}$) such that $x,y\in G$ implies that $x-y\in G$. Show that $G=\mathbb{R}$.
Clearly $0\in G$. Now how to show that all real numbers are there in $G$? Please help.

More generally, every additive subgroup $G$ of $\mathbb R$ is either discrete or dense (*). Since a discrete set in $\mathbb R$ is not open, $G$ must be dense. Now, the only open and dense subset of $\mathbb R$ is $\mathbb R$ itself.
(*) Consider $\alpha = \inf \{ x \in G : x>0 \}$. Then $G$ is discrete iff $\alpha >0$, in which case $G=\alpha \mathbb Z$. 
