The DCT states that if $f_n$ is a sequence of measurable functions and if there exist $g\in L$ such that $|f_n|\le g$ and $f_n\to f$ pointwise then $f\in L,$ $\lim\int|f_n-f|=0$ and $\lim\int f_n=\int f$
(Similar result to the $L^p$ version)
In this proof there is $|f-S_k(f)|^p\le2^{p+1}|g|^p.....(1)$
I think that is because we want something similar to $|f_n|\le g$.
In this case, $f_n=f-S_k(f)$?
On the other hand, $2^{p+1}$ should be in (1)?
How to remove to have only $|f-S_k(f)|^p\le |g|^p$ ?
EDIT: I changed the name of the sequence in the statement of the DCT to $h_k$ to avoid confusion with the sequences in the proof.
Yes, you're correct, they are defining $h_k$ by $h_k(x)=f(x)-S_k(x)$ (if you're using the $L^p$ version of the dominated convergence theorem; otherwise, take the $p$'th power of the absolute value). This goes to $0$ pointwise a.e. As for the dominating function, the constant $2^{p+1}$ doesn't affect integrability, so it doesn't matter. If you want to write it exactly in the form of the DCT (the standard version, similar for the $L^p$ version), you can take the dominator to be $G(x)=2^{p+1}|g(x)|^p.$ Since this is integrable, the dominated convergence theorem tells us that $$\|h_k\|_{L^p}=\|f-f_{n_{k+1}}\|_{L^p}\rightarrow 0,$$ and so $f_{n_k}\rightarrow f$ in $L^p$. The result now follows from Cauchiness.