I need to prove that the set $$S=\bigcap_{n=1}^\infty\bigcup_{q=2}^\infty\bigcup_{p=-\infty}^\infty\left(\frac{p}{q}-\frac{1}{q^n},\frac{p}{q}+\frac{1}{q^n}\right)-\left\{\frac{p}{q}\right\}$$ Is equal to the set $L$ of all Liouville numbers. It's easy to prove that $L\subset S$. Then, to prove that $S\subset L$ I take any $s\in S$, and then is easy to prove that $s$ satisfies all conditions to be a Liouville number, except one condition which I don't know how to prove, the condition that says that a Liouville number must be irrational. How do I prove that $s$ is not rational?
PS: Sorry for my bad english :(
