Showing an inequality in the proof of $L^p$ is a Banach space for $1\le p \lt \infty$.

Question

This is part of the proof that $L^p$ is a Banach space from Folland's Real Analysis, but there is a part that I don't understand.

Suppose $\{f_k\}\subset L^p$ and let $G_n=\sum_1^n |f_k|$ and $G=\sum_1^{\infty}|f_k|$ and $F=\sum_1^{\infty}f_k$.

Then, $|F-\sum_1^n f_k|^p \le (2G)^p \in L^1$.

My questions is how do we get this inequality $\le (2G)^p$? What I got is,

$|F-\sum_1^nf_k|^p=|\sum_{n+1}^\infty f_k|^p\le \sum_{n+1}^\infty |f_k|^p \le G^p$

I would greatly appreciate any help.

Answer 1

By triangle inequality, $$|F-\sum_1^n f_k|\le\left|\sum_1^\infty|f_k|+\sum_1^n|f_k|\right| \le 2G.$$

Your estimation $|\sum_{n+1}^\infty f_k|^p\le \sum_{n+1}^\infty |f_k|^p$ is not true. For example, $(1+2)^2=9>1^2+2^2=5$.