Prove the Axiom of Archimedes.

Let $x$ be a real number in $[0,1]$, and let $m, n$ be positive integers. Show that the limit

$$\lim _{m \rightarrow \infty}\left[\lim _{n \rightarrow \infty} \cos ^{2 n}(m ! \pi x)\right]$$

exists, and that its value depends on whether $x$ is rational or irrational.

[You may assume standard properties of the cosine function provided they are clearly stated.]