Show that $\lim_{k\to\infty}\lim_{j\to\infty}\cos^{2j}k!\pi x=0$

Question

The Dirichlet function is defined as the indicator function of rational numbers. I have also seen this function described by: $$f(x)=\lim_{k\to\infty}\lim_{j\to\infty}\cos^{2j}k!\pi x$$ How does this limit act as the indicator, and how does it yield an answer if cosine is limited to Infinity?

Answer 1

Suppose $x$ is rational and $x=\frac pq$ for $\gcd(p,q)=1$. If $k\ge q$, $k!x$ will be an integer, so $\cos k!\pi x=\pm1$ and $\cos^{2j}k!\pi x=1$. Thus the function is 1 for sufficiently large $k,j$, so it evaluates to 1.

Suppose $x$ is irrational, then $k!x$ will never be an integer, so $|\cos k!\pi x|<1$ for all $k$. As $j\to\infty$, this variable being in the exponent yields $\cos^{2j}k!\pi x\to0$, so the function evaluates to 0.