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Can we show that $$\sum_{n=0}^{\infty} \left(\int_n^{n+1} |f| dt\right)^p \leq \int_\mathbb{R}|f|^pdt $$ for $1\leq p\leq\infty$?

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    $\begingroup$ Should the integral be $\int_0^\infty$? $\endgroup$ – angryavian Mar 21 '18 at 21:55
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By writing $\int_0^\infty |f|^p \, dt = \sum_{n = 0}^\infty \int_n^{n+1} |f|^p \, dt$, it suffices to check $$\left(\int_n^{n+1} |f| \, dt\right)^p \le \int_n^{n+1} |f|^p\, dt.$$ When $p \ge 1$, this is a consequence of Jensen's inequality, since $x \mapsto x^p$ is convex.

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