Density of a real subset $A$ such that $\forall (a,b) \in A^2, \ \sqrt{ab} \in A$

Question

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!

Answer 1

Choose $a < b$ in $A$ (ignoring the trivial case where $A$ is a singleton), and consider the sequence with $a_0 = b$ and $a_{n+1} = \sqrt{a_n a}\in A$. It's decreasing and so must converge to $a$. Furthermore, since $a_{n+1} = a(b/a)^{1/2^n}$ with $a\not = b$, the sequence is eventually irrational.