Let $E$ be called dense in $\mathbb{R}$ if and only if $\text{int}(\mathbb{R} \setminus E)=\emptyset.$
Let $x \in\text{int}(\mathbb{R} \setminus E)=\emptyset$. Then for $\epsilon >0$, $(x-\epsilon, x+\epsilon)$ and $x \not\in E$. Hence $E$ is dense.
I'm not sure if this is the correct proof for the $\Leftarrow$ direction of the proof. Furthermore, I'm not sure how to proceed for the other direction of the proof.