I have two doubts on the initial statements on this proof of the completeness of $l^1$.
I don't understand why we can say that the sequence $x_k^{(n)}$ is a Cauchy sequence in $R$. I understand that $|x_k^{(n)} - x_k^{(m)}| \le \epsilon$, my doubt is that $x^{(n)}$ and $x^{(m)}$ are two different sequences.
When defining a Cauchy sequence in $R$ we say that a sequence is Cauchy if if for every positive real number $\epsilon_1$, there is a positive integer $N$ such that for all natural numbers $m, n > N$$$|x_m - x_n| < \varepsilon $$
In this case the distance is between elements of the same sequence.In the proof we have that the components of different sequences can be made arbitrary small. I hope I have made clear what is confusing me.
The other doubt is that I do not understand why we suspect $X$ to be the limit of the sequence $X_n$, what makes us suspect that?Maybe solving my previous doubt will help me understand this.