Factorial cosine iterated limit detects rationality (${\lim }_{m\to \infty }{\lim }_{n\to \infty }{\cos }^{2n}(m!\pi x)=1_{\mathbb{Q}}(x)$)

For fixed $m$, the inner limit is one precisely when $m!x$ is an integer, and zero otherwise, because $|\cos (\pi y)|=1$ exactly at integers and powers of a number with modulus below one tend to zero. If $x=p/q$ is a rational number, $q$ divides $m!$ for every $m\ge q$, making the inner limit eventually one. If $x$ is an irrational number, multiplication by a nonzero integer cannot make it integral, making every inner limit zero. The order of the two limits is essential to this argument.