In my notes we prove completeness of $L^p$ by showing that if $\sum\|f_k\|_p < \infty$ then $\sum f_k$ converges in $L^p$. (That's a lemma we prove a bit earlier, namely that $(V, \|\cdot\|)$ is Banach if and only if $\sum \|v_k\|< \infty$ implies $\sum v_k \to v \in V$.)
First we show that if $\sum\|f_k\|_p < \infty$ then $f(x) = \sum f_k(x)$ is in $\mathbb R$ $\mu$ almost everywhere. Once we have that, the notes do the following to show that $\sum_{k=1}^n f_k(x) \to f(x)$ in $\|\cdot\|_p$ (where $g(x) = \sum_{k=1}^\infty |f_k(x)|$):
My question: Do we really have to apply montone convergence yet again? I would do it like this, but maybe that's wrong so I'd appreciate your correction:
$ \|f(x) - \sum_{k=1}^n f_k(x)\|_p = \| \sum_{k=1}^\infty f_k(x) - \sum_{k=1}^n f_k(x)\|_p = \|\sum_{k=n+1}^\infty f_k(x)\|_p \stackrel{\Delta-\text{ineq.}}{\leq} \sum_{k=n+1}^\infty \|f_k \|_p $
Since by assumption $\sum \|f_k \|_p < \infty$we have that $\|f_k\|_p$ must be a null sequence (that is, it tends to $0$) hence for $n$ large enough, $ \sum_{k=n+1}^\infty \|f_k \|_p < \varepsilon$ and we're done.