Exercise 7, § 3

Let $E$ be a normed space over a valued division ring. Show that if every absolutely summable family in $E$ is summable in $E$, then $E$ is complete. [Let $(x_n)$ be a Cauchy sequence in $E$, and consider a subsequence $(x_{n_k})$ of $(x_n)$ such that the series whose general term is $x_{n_{k+1}} - x_{n_k}$ is absolutely convergent.]