x belongs to [0,1]
limit n->infinity (limit m-> infinity cos ^{2n} (m! pi x) )=?
its value is 0 if x is irrational
and it takes 1 if x is rational.
please explain how it happens.
I am not sure about this nested limit, but we get the Dirichlet function if we swap limits:
$\displaystyle \lim_{m\to\infty}\left(\lim_{n\to\infty}\left(\cos (m!\pi x)^{2n}\right)\right)$.
If x is rational, then m!x is eventually an (even) integer. If x is irrational, then $\displaystyle \cos(m!\pi x)\in(-1,1)$, so the inner limit is 0.