Let $a_n\geq a>0$, for any $x\in\Bbb R$, can we find a rational sequence $r_n$ such that $r_na_n\to x$?

Question

Let $a_n\geq a>0$, for any $x\in\Bbb R$, can we find a rational sequence $r_n$ such that $r_na_n\to x$?

My attempt: if $x/a_n$ is rational for any $n$, then $r_n=x/a_n$ is OK$. If $x/a_n$ is irrational, use Dirichlet? $|r_n-x/a_n|<\frac{C}{r_n^2}? Oh... $|r_na_n-x|<\frac{1}{n}$, $|a_n-\frac{x}{r_n}|<\cdots$...Oh.