'uniform approximation' of real in $[0,1]$ Good evening,
Prove that: For every $\varepsilon>0$, there exist an $n\in \mathbb{N}$, such that for every $x\in[0,1]$, there exist $(p,q)\in \mathbb{N^2}$, with $0\leq p\leq q\leq n$ and |$qx−p|≤\varepsilon$.
I have tried to prove the result like the proof of density but I didn't succeed.
Thanks for your help.

Hint: Look at Dirichlet's approximation theorem.
