Let $a_i$ be a sequence of real numbers and let p>1. Then:
$\sum_{i \in\mathbb{N}} |a_i|\leq (\sum_{i \in\mathbb{N}} |a_i|^p)^{1/p}$.
"Proof": $\sum_{i \in\mathbb{N}} |a_i|=((\sum_{i \in\mathbb{N}} |a_i|)^p)^{1/p}\leq (\sum_{i \in\mathbb{N}} |a_i|^p)^{1/p}$
where the last inequality holds by Jensen's inequality. This proof works for the case n=2, a case in which it clearly must be false.
You seem to be using $$\left(\sum_{i=1}^\infty|a_i|\right)^p\leq\sum_{i=1}^\infty|a_i|^p.$$ But this inequality holds in the opposite direction.