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?