I'd like to check if I understood the proof that $L^p$ is complete ($1 \le p <+\infty$).
I have to use the following fact: in a metric space, if a Cauchy sequence has a convergent subsequence then all the sequence is convergent (hence the space is complete).
Take $\{f_n\}$ a Cauchy sequence in $\mathcal L^p$ and construct a subsequence $\{f_{n_k}\}$ such that $\|f_{n_{k+1}}-f_{n_k}\|_p \le 1/2^k \;\;\forall k$
Define $g_r=\sum_{k=1}^r |f_{n_{k+1}}-f_{n_k}|$ and $g=\sum_{k=1}^\infty |f_{n_{k+1}}-f_{n_k}|$
For the Minkowski inequality $\|g_r\|_p\le\sum_{k=1}^r \|f_{n_{k+1}}-f_{n_k}\|_p\le\sum_{k=1}^\infty 1/2^k=1 \Rightarrow g_r \in \mathcal L^p \;\forall r$
Fo the MCT ($\sup_r \int g_r^p \le 1$)
$g_r\le g_{r+1} \; \forall r , \; g_r\to g \Rightarrow g_r^p\le g_{r+1}^p \; \forall r , \; g_r^p\to g^p \Rightarrow \int g^p=\lim_r \int g_r^p \le 1 \Rightarrow g \in \mathcal L^p$
It follows that $g<+\infty$ a.e., hence
$\sum_{k=1}^\infty |f_{n_{k+1}}-f_{n_k}|<+\infty$ a.e. $ \; \Rightarrow \; \sum_{k=1}^\infty (f_{n_{k+1}}-f_{n_k})<+\infty$ a.e. (*)
Defining $f=f_{n_1}+\sum_{k=1}^\infty (f_{n_{k+1}}-f_{n_k})$ where the series converges and putting $0$ elsewhere (over a null set), for construction (telescopic series) you have
$f=\lim_r[f_{n_1}+\sum_{k=1}^\infty (f_{n_{k+1}}-f_{n_k})]=\lim_r f_{n_{r+1}} \; \Rightarrow \; f_{n_k} \to f$ a.e.
Now by $|f| \le |f_{n_1}|+g$ you have $\int |f|^p \le \int |f_{n_1}|^p + \int g^p < + \infty \; \Rightarrow f \in \mathcal L^p$
Finally considering $h=|f_{n_k}-f|^p$ you have
$f_{n_k}-f \to 0$ a.e.$ \; |f_{n_k}-f|\le |f_{n_k}| + |f|\le 2(|f_{n_1}|+g) \Rightarrow h=|f_{n_k}-f|^p \to 0 $a.e.$ h=|f_{n_k}-f|^p \le 2^p(|f_{n_1}|+g)^p$
so by the DCT you can conclude
$\int h=\int |f_{n_k}-f|^p \to 0 \iff \; f_{n_k}\to f$ in $\mathcal L^p$.
It could be ok?
About the step (*), my idea is that it works for this argument (nothing special, the argument for series of functions with real domain is usual): a series converges absolutely a.e. means that the sequence of absolute values of partial sums converges a.e., then the sequence of partial sums is Cauchy in $\mathbb R$ so it converges.
Best Answer
The only mistake I see is where you claim that $|f|\leq|f_{n_1}|+|g|$ implies $|f|^p\leq|f_{n_1}|^p+|g|^p$ , which is not true ($1\leq1/2+1/2$ does not imply $1^2\leq (1/2)^2+(1/2)^2$). What you have to use is that the sum of two functions in $ L^p $ is again in $ L^p $.