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When $f_n$ if non-negative and increasing on $(0,\ \infty)$

$$\lim_{x\to \infty}\sum_{n=1}^{\infty}f_n(x)<\infty$$

Prove that $$\sum_{n=1}^{\infty}f_n'(x)<\infty$$ on $(0,\ 1)$ a.e $[m]$.

Is there the question means $f$ is differentiable? If so I will try mean value theorem.

If not, I am totally stuck at the beginning, since $f$ is not mentioned absolutely continuous or f' belong to $L^1(m)$, I have no idea how to connect $f'$ and $f$ here.

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    monotonic functions are differentiable almost everywhere.2012-11-29
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    What is $f_n$? $\ $2012-11-29

2 Answers 2