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Extending the triangle inequality, it's clear that $$\bigg| \sum_{i=1}^{n} a_{i} \bigg| \leq \sum_{i=1}^{n} |a_{i}|.$$ Is it true that $$\bigg| \sum_{i=1}^{n} a_{i} \bigg|^{p} \leq \sum_{i=1}^{n} |a_{i}|^{p},~~\textrm{for}~ p >1 ?$$ If so, how can I prove this ? Any help is much appreciated.

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    $\begingroup$ Deleting a question after an answer has been provided is an abusive behaviour. Please do not do it. $\endgroup$ – Jack D'Aurizio Apr 30 '18 at 23:06
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No. You could check for yourself: for $a_1,a_2 \geq 0$, do we always have $$(a_1+a_2)^2 \leq a_1^2+a_2^2$$ ? We don't (e.g. with $a_1=a_2=1$). So this fails even for $n=2$.

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