real analysis - How to prove the density of irrational numbers in $mathbb{R}$ without proving density of rationals first

I am asked to prove the density of irrationals in $\mathbb{R}$. I understand how to do this by proving the density of $\mathbb{Q}$ first, namely, adding a known irrational number such as $\sqrt{2}$ to $x,y \in \mathbb{R}$ ($xHowever, my professor has said that I can prove density of $\mathbb{R} \setminus \mathbb{Q}$ without even using the density of $\mathbb{Q}$ and it is a simple proof. I have puzzled over this for quite some time. I appreciate any help provided on this question in advance.