Double limit of $\cos^{2n}(m! \pi x)$ at rationals and irrationals

The intuition is already almost the proof.

• The cosine takes values between $-1$ and $1$, inclusive
• Taking an even power ensures that the values are between $0$ and $1$.
• Taking higher and higher powers will not affect those $x$ where the cosine squared is $1$, but all other values will converge to $0$

Thus $f(x):=\lim_{n\to\infty}\cos^{2n}(x)$ is a function that has value $0$ for all $x$, except that $f(x)=1$ if $\cos(x)=\pm1$, i.e. if $x$ is a multiple of $\pi$. After this, we see that multiplying the argument with $\pi$ is done to have the value $1$ at integers instead of multiples of $\pi$. Then the multiplication with $m!$ serves the purpose to obtain $1$ for any integer multiple of $\frac1{m!}$. As $m\to \infty$, this exhausts the rationals (and only them).

First notice that $-1\le\cos m!\pi x\le 1$ for all $m$ and $x$, so $0\le\cos^2m!\pi x\le 1$. Suppose that $0\le y\le 1$; then

$$\lim_{n\to\infty}y^n=\begin{cases} 0,&\text{if }y\ne 1\\ 1,&\text{if }y=1\;. \end{cases}\tag{1}$$

Now $\cos^2 m!\pi x=1$ if and only if $m!x$ is an integer. Say $m!x=a$ for some integer $a$; then $x=\frac{a}{m!}$, a rational number. Thus, $m!x$ is never an integer when $x$ is irrational, and therefore for all $m$ we have $\cos^2m!\pi x\ne 1$. $(1)$ then implies that $$\lim_{n\to\infty}\cos^{2n}m!\pi x=0$$ for all $m$, and of course in that case $$\lim_{m\to\infty}\lim_{n\to\infty}\cos^{2n}m!\pi x=\lim_{m\to\infty}0=0\;.$$

This explains why the limit is $0$ when $x$ is irrational.

If $x$ is rational, on the other hand, we can write $x=\frac{a}b$ for some integers $a,b$ with $b>0$. Then for all $m\ge b$ we can be sure that $m!x$ is an integer and hence that $\cos^2m!\pi x=1$. But then the sequence

$$\left\langle\cos^{2n}m!\pi x:n\in\Bbb N\right\rangle$$

is constantly $1$ for $m\ge b!$, so for $m\ge b!$ we have

$$\lim_{n\to\infty}\cos^{2n}m!\pi x=1\;.$$

But then the sequence $$\left\langle\lim_{n\to\infty}\cos^{2n}m!\pi x:m\in\Bbb N\right\rangle$$

is constantly $1$ for $m\ge b!$, and

$$\lim_{m\to\infty}\lim_{n\to\infty}\cos^{2n}m!\pi x=1\;.$$

This is known as Dirichlet Function, If $m!x$ is an integer, then $\cos^{2n}(m!\pi x) = 1$.

If $x$ is rational $\frac{p}{q}$, then, eventually, for large enough $m$, $m!$ will be divisible by $q$ (and stay so for all larger m), so that $m!x$ will be an integer, and we have what we want.

If $x$ is irrational, $m!x$ will never be an integer, and $|\cos(m!\pi x)| \lt 1$, so that $\lim_{n \to \infty} \cos^{2n}(m!\pi x) = 0 $ for all $m>0$.