Suppose that $A$ is a non empty subset of the positive reals such that $$\forall (a,b) \in A^2, \ \sqrt{ab} \in A \tag{*}$$

How to prove that $A \cap (\mathbb R \setminus \mathbb Q)$ is dense in $(\inf A, \sup A)$?

*I'm trying to find sequences of irrational numbers belonging to $A$... without great success up to now!*