A real number $x$ is said to be dyadic rational provided there is an integer $k$ and a non negative integer $n$ for which $x=\frac{k}{2^n}$. For each $x \in [0, 1]$ and each $n \in \mathbb{N}$ set:
$$f_n(x) = \left\{\begin{matrix}
1 &, & x =\frac{k}{2^n} , \; k \in \mathbb{N} \\
0& , & \text{otherwise}
\end{matrix}\right.$$
- Prove that the dyadic numbers are dense in $\mathbb{R}$.
- Let $f:[0, 1] \rightarrow \mathbb{R}$ be the function to which the sequence $\{f_n\}_{n \in \mathbb{N}}$ converges pointwise. Show that $\bigintsss_0^1 f(x) \, {\rm d}x$ does not exist.
- Show that the convergence of $f_n \rightarrow f$ is not uniform.