Exercise 5, § 1

Let E be a metrizable vector space over a non-discrete valued field K. Prove that if $(B_n)$ is an arbitrary sequence of bounded subsets of E (III, p. 37, exerc. 1), then there exists a sequence $(\lambda_n)$ of scalars $\neq 0$, such that the union of the sets $\lambda_n B_n$ is bounded.