Let $A_n$ be such a subset of $\mathbb{R}$ such that for any $\epsilon>0$, if $x\in A_n$ then $x\in (\frac{p}{q}-\frac{\epsilon}{q^n}, \frac{p}{q}+\frac{\epsilon}{q^n})$ for some rational $\frac{p}{q}, \text{gcd}(p,q)=1$.
Prove that $\cap_n A_n-\mathbb{Q}$ is a dense subset of $\mathbb{R}$.
I'm wondering if Dirichlet's approximation theorem helps here, and I have tried to prove $\forall x\in \mathbb{R}, \delta>0$, there exists a number in $\cap_n A_n-\mathbb{Q}$ in $(x-\delta, x+\delta)$, but I can only prove this for $x\in \mathbb{Q}$. Thanks for any help.