Dual limit

As $x$ is irrational , $x\pi$ can never be equal to an integral multiple of $\pi$ . Hence $n!x\pi$ can also never equal an integral multiple of $\pi$ .

So $\cos(n!x\pi)$ can never equal $1$ or $-1$ . So $-1<\cos(n!x\pi)<1$ . So $0<\cos^{2m}(n!x\pi)<1$ . So $\displaystyle 0<\lim_{n\to\infty}\cos^{2m}(n!x\pi)<1$

Now as we know that for a real number $y\in(0,1)$ , $\displaystyle \lim_{m\to\infty} y^{2m} = 0$ .

Hence $\displaystyle \lim_{m\to\infty}\lim_{n\to\infty}\cos^{2m}(n!x\pi) = 0$

Hence $\displaystyle \lim_{m\to\infty}\lim_{n\to\infty} (1+\cos^{2m}(n!x\pi)) = 1$

Here it is needed to be mentioned that $n$ and $m$ are integers . Otherwise the result would be different.