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I am working through a set of exercises where I have just proven that a normed vector space is a Banach space iff every Cauchy sequence which is an absolutely convergent series is a convergent series. The next part of the question asks me to prove that $l^1$ is a complete vector space, but I have no idea how to porve it satisfies the equivalent condition. Help would be appreciated!

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Hint: Let$$\sum_{n=1}^\infty\left(\sum_{k=1}^\infty x_{nk}\right)\tag1$$be a series of elements of $\ell^1$ which is both a Cauchy sequence (meaning that the sequence of partial sums is a Cauchy sequence) and an absolutely convergent series (meaning that the series $\sum_{n=1}^\infty\left\|\sum_{k=1}^\infty x_{nk}\right\|_1$ converges). Then prove that:

  1. For each $n\in\mathbb N$, the series $\sum_{k=1}^\infty x_{nk}$ converges; let $s_k$ be its sum.
  2. The series$$\sum_{n=1}^\infty s_n\tag2$$belongs to $\ell^1$.
  3. The sum of the series $(1)$ is the series $(2)$.
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