Let $x$ be an irrational number in $[0, 1]$. Find $\lim \sup \cos (2\pi nx)$
What I think I need to do is in some way take an $\epsilon$ neighborhood around some arbitrary point close to $1$ and then show explicitly that the least upper bound is $1$ since the sequence in question gets arbitrary close to $1$.
What I need help with is setting up the necessary details to show what the lim sup is.