# For the sake of completeness

Let’s prove the completeness of $\ell^p$. The argument consists of two steps.

Claim 1. Suppose $X$ is a normed space in which every absolutely convergent series converges; that is, $\sum_{n=1}^{\infty} x_n$ converges whenever $x_n\in X$ are such that $\sum_{n=1}^{\infty} \|x_n\|$ converges. Then the space is complete.

Proof. Take a Cauchy sequence $\{y_n\}\subset X$. For $j=1,2,\dots$ find an integer $n_j$ such that $\|y_n-y_m\|<2^{-j}$ as long as $n,m\ge n_j$. (This is possible because the sequence is Cauchy.) Also let $n_0=1$ and consider the series $\sum_{j=1}^{\infty} (y_{n_{j}}-y_{n_{j-1}})$. By the hypothesis this series converges. Its partial sums simplify (telescope) to $y_{n_j}-y_1$. Hence the subsequence $\{y_{n_j}\}$ has a limit. It remains to apply a general theorem about metric spaces: if a Cauchy sequence has a convergent subsequence, then the entire sequence converges. This proves Claim 1.

Claim 2. Every absolutely convergent series in $\ell^p$ converges.

Proof. The elements of $\ell^p$ are functions from $\mathbb N$ to $\mathbb C$, so let’s write them as such: $f_j\colon \mathbb N\to \mathbb C$. (This avoids confusion of indices.) Suppose the series $\sum_{j=1}^{\infty} \|f_j\|$ converges. Then for any $n$ the series $\sum_{j=1}^{\infty} |f_j(n)|$ also converges, by Comparison Test. Hence $\sum_{j=1}^{\infty} f_j(n)$ converges (absolutely convergent implies convergent for series of real or complex numbers). Let $f(n) = \sum_{j=1}^{\infty} f_j(n)$. So far the convergence is only pointwise, so we are not done. We still have to show that the series converges in $\ell^p$, that is, its tails have small $\ell^2$ norm: $\sum_{n=1}^\infty |\sum_{j=k}^{\infty} f_j(n)|^p \to 0$ as $k\to\infty$.

What we need now is a dominating function, so that we can apply the Dominated Convergence Theorem. Namely, we need a function $g\colon \mathbb N\to [0,\infty)$ such that
(1) $\sum_{n=1}^{\infty} g(n)<\infty$, and
(2) $|\sum_{j=k}^{\infty} f_j(n)|^p \le g(n)$ for all $k,n$.

Set $g=(\sum_{j=1}^{\infty} |f_j|)^p$. Then (2) follows from the triangle inequality. Also, $g$ is the increasing limit of functions $g_k =(\sum_{j=1}^k |f_j|)^p$, for which we have
$\sum_n g_k(n) \le (\sum_{j=1}^k \|f_j\|)^p \le (\sum_{j=1}^{\infty} \|f_j\|)^p<\infty$
using the triangle inequality in $\ell^p$. Therefore, $\sum_n g(n)<\infty$ by the Monotone Convergence Theorem.

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