I have just proved the density of the rationals via the classic Archimedean Property method. I now need to prove the density of the irrationals.
I thought of a method which uses the density of the rationals to do this but I am not sure if it is valid.
Proof:
From the density of the rationals I have that $\forall x,y \in \mathbb{R},$$ \exists z \in \mathbb{Q}$ s.t. $x<z<y$
Since $x$ and $y$ can be any real value I set them to be $x + \sqrt2$ and $y + \sqrt2$. Then using the density of the rationals I can do the following $x + \sqrt2 < z < y +\sqrt2 \implies x<z-\sqrt2<y$ which yields an irrational number between $x$ and $y$.
Is this correct?