Dyadic rationals and more ...

Real Analysis
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Riemann
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Dyadic rationals and more ...

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Post by Riemann »

Inspired by this question

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:

$$f_n(x) = \left\{\begin{matrix}
1 &, & x =\frac{k}{2^n} , \; k \in \mathbb{N} \\
0& , & \text{otherwise}
\end{matrix}\right.$$
  1. Prove that the dyadic numbers are dense in $\mathbb{R}$.
  2. 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.
  3. Show that the convergence of $f_n \rightarrow f$ is not uniform.
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$
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