I am wondering why does the typewriter sequence defined here on example 4 does not converge pointwise?
$f_n= \mathbb{1}_{\left [\frac{n-2^k}{2^k},\frac{n-2^k+1}{2^k}\right]}$ for $ k\geq0 $ and $2^k \leq n < 2^{k +1}$
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1I think this will help you https://math.stackexchange.com/questions/1412091/the-typewriter-sequence – Javi Jan 19 '18 at 15:02
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Hint
For $x \in [0,1]$ with binary representation $x_0 x_1 x_2 \dots$ what are the values of $$f_{\overline{x_0 x_1 \dots x_{k-1}0}}(x), \ f_{\overline{x_0 x_1 \dots x_{k-1}1}}(x)?$$
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$n=\overline{x_0 x_1 \dots x_k}$ is the integer having for binary representation the "digits" $x_0, x_1, \dots, x_k$ so $n = \sum_{l=0}^k x_l 2^{k-l}$. – mathcounterexamples.net Jan 19 '18 at 15:26