Understanding a subsequence converges in the proof of completeness of $L^p$ in Stein-Shakarchi's Functional Analysis

Question

I am trying to understand the following proof of completeness of $L^p$ in Stein-Shakarchi's Functional Analysis. (See the proof on page 5 of the link or at the end of this post.)

In the proof, it is said that:

Since (by construction of the telescopic series) the $(K − 1)^{\mathrm{th}}$ partial sum of this series is precisely $f_{n_K}$ , we find that $f_{n_K}\to f$ a.e.

Question 1: why is this true?

What I tried is $f(x)=f_{n_1}(x)+f_{n_{K+1}}-f_{n_1}(x)+(f_{n_{K+2}}+f_{n_{K+3}}+...)=f_{n_{K+1}}+(f_{n_{K+2}}+f_{n_{K+3}}+...)$

If $K\to\infty$ is it correct to conclude $f_{n_K}\to f$ ?

Question 2: why $2^p|f(x)|^p+2^p|S_K(f)(x)|^p\le2^{p+1}|g(x)|^p,$ for all $K$ ?

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Answer 1

This is Theorem 1.3 in Section 1.2 (page 5) of Stein-Shakarchi's Functional Analysis.

Note that

• the $(K-1)^{\mathrm{th}}$ partial sum in the proof refers to $S_K(f)(x)$, which by the definition formula equals to $f_{n_K}$;
• by the proof, $S_K(f)\to f$ almost everywhere.

Thus the authors conclude that $f_{n_K}=S_{K-1}(f)\to f$ almost everywhere.

For the second question, simply note that by the triangle inequality, $$ |f(x)|\leq |g(x)|,\quad |S_K(f)|\leq |g(x)|. $$