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.

Use the following facts:


*

*Since $G$ is a subgroup hence $0 \in G$.

*Since $G$ is open there is an $\epsilon >0$ such that $(-\epsilon,\epsilon) \subset G$.
