Let $A$ be a normed space where every absolutely convergent series converges in $A$.
How do I prove that $A$ is Banach?
Let $\sum_{n=1}^\infty x_n$ be an absolutely convergent series in $A$, then we know that $\sum_{n=1}^\infty ||x_n||$ is convergent in $\mathbb{R}$ and that $\sum_{n=1}^\infty x_n$ is convergent in $A$.
Now using this, we want to show that all Cauchy sequences $\{x_n\}$ in $A$ converge. How would I do this?
Let $(x_n)_{n\geq 1}$ be Cauchy in $A$. We may extract a subsequence $x_{n_k}$ for which $\|x_{n_{k+1}}-x_{n_{k}}\|\leq 2^{-k}$ for $k\geq 1$. Then $$ \sum_{k\geq 1} \|x_{n_{k+1}}-x_{n_{k}}\| \leq \sum_{k\geq 1} 2^{-k} =1$$ implies by the hypothesis on $A$ that the limit $$y= x_{n_1} + \lim_{p\rightarrow \infty}\sum_{k=1}^p (x_{n_{k+1}} - x_{n_k})\in A$$ exists (as the sum is abs convergent). We have $\|y-x_{n_k}\|\leq 2^{1-k}$ so $x_{n_k}$ converges to $y$ in $A$ and then so does $x_n$.