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I have stumbled upon this problem which keeps me from finishing a proof:

$(\sum_{n} {|X_n|})^a \leq \sum_{n} {|X_n|}^a$, where $n \in \mathbb{N}$ and $ 0 \leq a \leq 1 $

I have no idea how to prove this. It is something like the Cauchy-Schwarz inequality which applies in case $0 \leq a \leq 1$?

Any tip is welcome. Thanks!

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    I have tried starting with the Minkovsky inequality..but I don't seem to get anywhere.2012-11-28

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