Exercise 16, § 1

Let E be a locally convex metrizable space satisfying the first axiom of countability, such that its completion $\hat{E}$ is a Fréchet space which satisfies the first axiom of countability.

Show that every bounded subset B of $\hat{E}$ is contained in the closure of a bounded subset of $\hat{E}$. (Reduce to the case where B is countable, arranged as a sequence $(x_n)$; on the other hand, let $(p_n)$ be an increasing sequence of semi-norms defining the topology of $\hat{E}$; for each integer n consider a sequence $(y_{nk})_{k \geq 1}$ of points of E, which converges to $x_n$ and is such that $p_n(x_n - y_{nk}) \leq 1$ for all $k \geq 1$.)