We know that if $x,y\in\mathbb{R}$, $x<y$ then exists $q\in\mathbb{Q}$ such that $x<q<y.$ Now, we consider $\overline{\mathbb{R}}=\mathbb{R}\cup\{\pm\infty\}.$ We exetend the order relation of $\mathbb{R}$ on $\overline{\mathbb{R}}$ In the following way $$-\infty<x<+\infty\quad\text{for all}\quad x\in\mathbb{R}.$$
Question. Based on what I said can I say that for all $x,y\in\overline{\mathbb{R}}$ $x<y$ exists $q\in\mathbb{Q}$ such that $x<q<y$?
Thanks!