Characteristic function of the rationals equals the double limit of continuous functinos

Question

Let $\chi_{\mathbb{Q}} $. why does it follow that

$$ \chi_{\mathbb{Q}} = \lim_{n \to \infty} \lim_{m \to \infty} \big( \cos(2 \pi n! x)\big)^{2m}$$

Answer 1

First note $\lim_{m\to\infty}\cos^{2m} x = 1$ if $x = k\pi$ for integer $k$ and $\lim_{m\to\infty}\cos^{2m} x = 0$ otherwise.

Let

$$f_n(x) = \lim_{m\to\infty} \left( \cos(2\pi n! x) \right)^{2m}.$$

Then given a rational number $r = p/q$, for all $n \geq q$ we have $f_n(r) = 1$ and hence $\lim_{n\to\infty} f_n(r) = 1$.

The last thing you want to convince yourself of is that if $x$ is not rational then $f_n(x) = 0$ for all $n$. In which case we have the pointwise limit, $\chi_{\mathbb{Q}}(x) = \lim_{n\to\infty} f_n(x)$ for all real numbers $x$.