I am trying to prove that $f(n, m)=|cos(n)-cos(m)|$ where $n$, $m$ are distinct natural numbers, can be made as small as possible; or that the greatest lower bound on $f(n, m)$ is $0$.
I think the answer lies in proving that for any irrational number $r$, the greatest lower bound of $|n-mr|$ is $0$. I can't think of anyway to proceed further. I have only ever dealt with ranges of single variable functions so this is a real roadblock for me. Please help me find a way.