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Why is it that $\Big|\sum_{k=0}^{N}\int_{0}^{\pi}\frac{\sin(y)}{(k+1)\pi}dy\Big|\leq\sum_{k=0}^{N}\int_{k\pi}^{(k+1)\pi}\frac{|\sin(y)|}{y}dy$

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I mean, I know that it is true because by graphing each of the functions for each $N$ gives me that the difference is as small as I want. But why is that? How can I answer to this question? I think this has to do with the fact that $\int_{k\pi}^{(k+1)\pi}\frac{|\sin(y)|}{y}dy$ is a monotone function, but I don't really know why. Anyone has a hint? Thank you.


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