I would like to show that $\ell^p$, for $p \in [1, \infty)$ is Banach, using the following fact: Given a normed linear space $X$, $X$ is Banach if and only if absolute convergence in $X$ implies conditional convergence in $X$. Here's what I've done:
Let $\{ x_n \}$ be a sequence in $\ell^p$ with the property that $\sum_{n=1}^\infty\| x_n \|_{\ell^p} < \infty$. I must show that $\sum_{n=1}^\infty x_n := \lim_{N \to \infty} \sum_{n=1}^N x_n$ is an element of $\ell_p$. But we have the chain of inequalities
\begin{align} & \sum_{\ell=1}^\infty \Bigl| \Bigl( \sum_{n=1}^\infty x_n \Bigr)(\ell) \Bigr|^p = \sum_{\ell=1}^\infty \Bigl| \lim_{N \to \infty}\sum_{n=1}^N x_n (\ell) \Bigr|^p \leq \sum_{\ell=1}^\infty \lim_{N \to \infty} \sum_{n=1}^N \Bigl| x_n (\ell) \Bigr|^p \\[8pt] = {} & \sum_{\ell=1}^\infty \sum_{n=1}^\infty \Bigl| x_n (\ell) \Bigr|^p = \sum_{n=1}^\infty \sum_{\ell=1}^\infty \Bigl| x_n (\ell) \Bigr|^p = \sum_{n=1}^\infty \|x_n\|^p_{\ell^p} < \infty, \end{align}
where the final inequality holds because $\sum_{n=1}^\infty\| x_n \|_{\ell^p} < \infty$. Thus $\sum_{n=1}^\infty x_n \in \ell^p$, as desired.
Is my proof correct? If not, what can I do to fix it?
You can use Minkowski's inequality to repair the proof. First of all, note that for every $k$ the numeric series $\sum_n x_n(k)$ converges absolutely; let $x(k)$ denote its sum. The goal is to prove that $x\in \ell^p$. By Minkowski's inequality, the partial sums $S_N=\sum_{n=1}^Nx_n$ are uniformly bounded in $\ell^p$: $$ \|S_N\|_p=\left\| \sum_{n=1}^Nx_n \right\|_p\le \sum_{n=1}^N\|x_n\|_p \le \sum_{n=1}^\infty \|x_n\|_p $$ Fatou's lemma implies $$ \sum_k |x(k)|^p \le \liminf_{N\to\infty} \sum_k |S_N(k)|^p \tag1$$ which is finite by the above.
(Fatou's lemma is being applied to the measure space $\mathbb{N}$ with the counting measure. If you don't like this, prove (1) directly by considering partial sums over $1\le k\le K$.)