Prove that every absolutely summable series in L1 is summable

Question

I'm working through a real analysis book to study for my qualifying exams, and came across this problem. Let $\ell_1$ be the set of all real sequences such that $\sum_{n=1}^{\infty}\lvert x_n \rvert < \infty$. Let $x_n^k$ be a sequence in $\ell_1$ Show that if $\sum_{k=1}^{\infty}\lVert x_n^k\rVert < \infty$ then $\sum_{k=1}^{\infty}x_n^k$ converges to some $x_n \in \ell_1$. I don't know where to start, and also don't use that $\ell_1$ is complete because the problem actually states to prove that $\ell_1$ is complete by showing that every absolutely summable series is summable.

Answer 1

Hints: let $\epsilon >0$ and choose $N$ such that $ \sum\limits_{k=N+1}^{\infty}\|x_n^{k}\| <\epsilon$. Then show that $\sum\limits_{k=N_1}^{N_2} |x_n^{k}| <\epsilon$ whenever $N_2>N_1>N$. Conclude that the partial sums of the series $\sum\limits_{k=1}^{\infty} x_n^{k}$ form a Cauchy sequence so $x_n =\sum\limits_{k=1}^{\infty} x_n^{k}$ exists. Now can you verify that $\sum\limits_{k=1}^{\infty} |x_n| \leq \sum\limits_{k=1}^{\infty} \|x_n^{k}\|$?