Computing double limit of $\lim_{n\to\infty} \lim_{m\to\infty} \cos^m(n!2\pi x)$

Question

Compute, for every real value $x$, the double limit:

$$\lim_{n \to \infty} \lim_{m \to \infty} \cos^m(n!2\pi x)$$

How do i go about solving the above problem? Note that this is a problem from a real analysis course, not calculus.

Thanks for looking at my question.