I have the following statement
"If $x$ and $y$ are real numbers with $x < y$, then there exists an irrational number $z$ such that $x < z < y$.
So, I have begun by applying the density theorem that there exists a rational number $r$ such that $x < r < y$. The theorem also states that $r$ cannot be equal to $0$ but I am not sure why. Does it have something to do with the Archimedean Property?
Any help would be appreciated.