Proof that $L^{p}$ is complete in Folland's Real Analysis

Theorem $5.1$ says that a normed space is complete iff, whenever $\sum_{n=1}^\infty \|x_n\|$ converges, then $\sum_{n=1}^\infty x_n$ converges (in the norm, of course). Folland's wording is: "A normed vector space $X$ is complete iff every absolutely convergent series in $X$ converges." However this is the same as what I've written above, because he defines that a series $\sum_{n=1}^\infty x_n$ converges to $x$ if $\sum_{n=1}^N x_n\to x$ as $N\to\infty$, and that a series $\sum_{n=1}^\infty x_n$ is absolutely convergent if $\sum_{n=1}^\infty\|x_n\|<\infty$.

To apply this theorem, we assume that $\sum_{k=1}^\infty \|f_k\|_p$ converges and must show that $\sum_{k=1}^\infty f_k$ converges in the $L^p$ norm. This is precisely what Folland does.

It suffices to check that every absolutely convergent series is convergent. Suppose, $$\sum_{k=1}^{\infty}\|f_k\|_{p} = B < \infty$$ We need $g\in L^p$ so that $$\lim_{n\to \infty}\|g - \sum_{k=1}^{n}f_k\|_{p} = 0$$ Set $G = \sum_{k=1}^{\infty}|f_k|$ and $G_n = \sum_{k=1}^{n}|f_k|$. Then by the triangle inequality for $L^p$, $$\|G_n\|_{p} \leq \sum_{k=1}^{n}\|f_k\|_{p}\leq B$$ by the Monotone Convergence Theorem, $$\|G\|_{p} \leq B$$ so $G^p(x) < \infty$ for a.e. $x$ so $G(x) < \infty$ for a.e. $x$. Thus $g = \sum_{1}^{\infty}f_k$ is absolutely convergent for a.e. $x$ so convergent a.e. Since $\|g\|\leq G$ we have $g\in L^p$. Now we need to show that $$\lim_{n\to \infty}\int |g - \sum_{k=1}^{n}f_k|^p d\mu = 0$$ We have $$|g(x) - \sum_{1}^{n}f_k(x)|^p\leq (|g(x)|+ \sum_{1}^{n}|f_k(x)| )^p \leq (G+G)^p=(2G)^p\in L^1$$ By the Dominated Convergence Theorem, $$\lim_{n\to \infty}\int |g - \sum_{k=1}^{n}f_k|^p d\mu = 0$$ Thus the series $\sum_{k=1}^{\infty}f_k$ converges in $L^p$.