Showing that $\lim \int \left(\sum_1^n |f_k|\right)^p \le \left(\sum_1^\infty \|f_k\|_p\right)^p$

Question

I am reviewing a proof about the completeness of $L^p$ spaces. The proof begins as such (Folland Theorem 6.6):

For $1 \le p < \infty$, suppose $\{f_k\} \subset L^p$ and $\sum_1^\infty \|f_k\| = B < \infty$. Let $G_n = \sum_1^n |f_k|$ and $G = \sum_1^\infty |f_k|$. Then $\|G_n\|_p \le \sum_1^n \|f_k\|_p \le B$ for all $n$, so by the monotone convergence theorem, $\int G^p = \lim \int G_n^p \le B^p$....

Expanding the second part of the final relation quoted above, we get

$$\lim \int G_n^p = \lim \int \left(\sum_1^n |f_k|\right)^p \le \left(\sum_1^\infty \|f_k\|_p\right)^p.$$

My question: How do we obtain the inequality in the above?

Answer 1

From Minkowski, $ \| \sum_1^n |f_k|\|_p \le \sum_1^n \|f_k\|_p \le \sum_1^\infty \|f_k\|_p.$ Now raise to the $p$th power.

Answer 2

Triangle inequality then dominated convergence theorem.

Edit: By triangle inequality, we have, $$\int G^p= \lim \int G_k^p=\lim \int (\sum_{k=1}^n |f_k|)^p=\lim ||(\sum_{k=1}^n |f_k|) ||^2\leq \lim \sum_{k=1}^n ||f_k||^2$$ By DCT, we then conclude the given inequality.