Prove the following:

$\displaystyle \lim_{n\to\infty}\lim_{k\to\infty}\cos^{2k}(\pi n! x)=\begin{cases} 1 \text{ if }x \in\mathbb{Q} \\ 0 \text{ if } x\in \mathbb{R}/\mathbb{Q}\end{cases}$ and show that:

$\displaystyle \lim_{n\to\infty}\lim_{k\to\infty}\cos^{2k}(\pi n! e)\neq \lim_{k\to\infty}\lim_{n\to\infty}\cos^{2k}(\pi n! e)$