I am using the theorem :
An NLS $V$ is complete if and only if every absolutely summable series converges in $V$
Let $\sum x^n$ be absolutely summable where $x^n:=(\xi^n_1,\xi^n_2,...)$. Let $s^n:=x^1+x^2+...+x^n$ be the $n^{th}$ partial sum of the series. To show: {$s^n$} converges in $l_∞$
We have $s^n=(\xi^1_1+\xi^2_1+...+\xi^n_1,\xi^1_2+\xi^2_2+...+\xi^n_2,...)$ Since $\sum x^n$ is absolutely summable, we have $\sum ||x^n||_∞<∞$ i.e. $\sum \sup_i|\xi^n_i|<∞ \implies \sum |\xi^n_i|<∞ \forall i$. So $\sum \xi^n_i$ converges in $\Bbb R$, to say, $s_i$. Indeed $s^k\rightarrow s:=(s_1,s_2,...)$ because $||s^k-s||=\sup_i |\sum \xi^{k+n}_i|$, which I can make as small as I please by taking large enough $k$ since $\sum \xi^{n}_i$ is finite. $||s||_∞=\sup_i |\sum \xi^n_i|≤\sup_i\sum |\xi^n_i|<\sum \sup_i |\xi^n_i|<∞$.
Is this proof acceptable? This is the first time I am using this theorem so I'm not sure if I applied it correctly.