By struggling with the proof that $\ell^p$ is complete, I looked up different proofs by different authors, and I ended up focusing on the one given by Kreyszig in his classic book on functional analysis, because I found it the most perspicuous for my level. Still, there are some point that are not completely clear, thus I will write down here the whole proof and I will add my remarks and doubts.
[I]
Theorem: The space $\ell^p$ is complete; here $p$ is fixed and $1 \leq p < \infty$.
Proof: Let $(x_n)$ be any Cauchy sequence in $\ell^p$, where $x_m = ( \xi^{(m)}_1, \xi^{(m)}_2, \dots )$. Then, for every $\varepsilon > 0 $ there is an $N$ such that for all $m, n >N$, \begin{equation} d ( x_m, x_n ) = \Bigg( \sum_{j=1}^\infty |\xi^{(m)}_j - \xi^{(n)}_j |^p \Bigg)^{\frac{1}{p}} < \varepsilon \hspace{2cm} \text{(1)} . \end{equation} It follows that for every $j = 1,2, \dots$ we have for $m,n >N$ $$ |\xi^{(m)}_j - \xi^{(n)}_j | < \varepsilon \hspace{2cm} \text{(2)}. $$ We choose a fixed $j$. From (2) we see that $( \xi^{(1)}_j, \xi^{(2)}_j, \dots )$ is a Cauchy sequence of numbers. It converges since $\Re$ is complete, say $\xi^{(m)}_j \to \xi_j$ as $m \to \infty$. Using these limits, we define $x = (\xi_1, \xi_2, \dots)$ and show that $x \in \ell^p$ and $x_m \to x$.
This looks fine,
[II]
From (1) we have for all $m,n >N$ $$ \sum_{j=1}^k |\xi^{(m)}_j - \xi^{(n)}_j |^p < \varepsilon^p \hspace{2cm} (k=1,2,\dots). $$
First problems!
I see where the $\varepsilon^p$ comes from, but what puzzles me is the $k$ on the top of $\sum$. I see it comes from (1) (indeed, if it works for $j \to \infty$, then it has to work necessarily for a finite $k$. Still, I don't see why we actually need to do it.
Why don't we simply stick to (1)?
[III]
Letting $n \to \infty$, we obtain for $m>N$ $$ \sum_{j=1}^k |\xi^{(m)}_j - \xi_j |^p \leq \varepsilon^p \hspace{2cm} (k=1,2,\dots). $$
Same problem as before, plus the mysterious $\leq$ between LHS and RHS. Indeed, I think the idea should be that the sequence converge and it is Cauchy, thus the $\varepsilon > 0$ is the same that we use in the definition of Cauchy sequence, and in the standard limit definition. But then it should be still $<$ and not $\leq$.
[IV]
We may now let $k \to \infty$; then for $m >N$ $$ \sum_{j=1}^\infty |\xi^{(m)}_j - \xi_j |^p \leq \varepsilon^p \hspace{2cm} \text{(3)}. $$
Bypassing the previous related problems, this is fine.
[V]
This shows that $x_m - x = ( \xi^{(m)}_j - \xi_j ) \in \ell^p$.
Not completely sure I see why it is actually the case.
[VI]
Since $x_m \in \ell^p$, it follows by means of the Minkowski inequality, that $$ x = x_m + (x - x_m) \in \ell^p. $$
The reference to the Minkowski inequality is really mysterious.
[VII]
Furthermore, the series in (3) represents $[d(x_m,x_)]^p$, so that (3) implies that $x_m \to x$. Since $(x_m)$ was an arbitrary Cauchy sequence in $\ell^p$, this proves the completeness of $\ell^p$, where $1 \leq p < \infty$. QED
Sorry for the vivisection of this rather straightforward proof, but I have the feeling that by properly catching each step here, I could improve dramatically my overall understanding of real analysis tools and procedures.
Thank you for your time and patience. As always, any feedback is most welcome!
Edit for BOUNTY:
I am editing this question because I am putting a bounty. True enough, I received very helpful comments, but still they were only comments, and I would really love to have an answer, because I do feel a lot of the things that look here mysterious have to be fairly important.
I also have the feeling that some of the problems I showed in these questions, e.g. the one that is below under (1), can give the feeling that it is impossible I can actually try to read something sort of advanced, without having a solid grasps of other things, and maybe this could put a potential answerer in the akward position of feeling “Come on, man, are you kidding me? Don’t let me waste my time. I cannot go back to teach you 2+2!”. However, that’s how things are, and this is mostly due to the fact that I am self-taught, which put me in the position to choose my own topics. But, actually, exactly those very naive questions are the ones that would set me on the right path to keep on studying properly.
Hence, I renumbered the parts in which I divided the proof of the theorem in order to easily refer the questions to each part.
1) Is the change from $\infty$ to $k$ in [II] related to the fact that we are dealing with a series, and thus we standardly see how a series behave with its finite terms, before letting the limit goes to $\infty$ (which is what happen in [III]?
2) Is in step [V] implicitly assumed that it is the case due to the fact that it is for all $j$?
3) How does step [VI] come from the Minkowski inequality?
Of course, any other addition or explanation is most welcome.
Thanks for your time!