I was studying for my analysis mid-term paper and was going over the properties of real numbers. I was wondering how to prove the following statement: (Not a textbook problem, it just popped into my head.)
Given rational numbers $p$ and $q$ such that $p < q$, show that there exists an irrational number $r$ such that $p < r < q$.
I know some ways of proving it, like picking a known irrational and shifting it into the open interval $(p,q)$. I was wondering whether there is a way to prove it without referencing to any previously known irrationals. Specifically I am trying to construct a sequence of rational numbers which converges to a irrational in the interval $(p,q)$. Is there any way to do that?