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!