Proposition 2

A locally convex metrizable space is bornological.

Suppose E is metrizable, and p a semi-norm on E which is bounded on bounded subsets of E, but which is not continuous. Let A be the set of all $x \in E$ such that $p(x) < 1$. Let $(V_n)_{n \geq 1}$ be a decreasing sequence forming a fundamental system of neighbourhoods of 0 in E. Since p is not continuous, A is not a neighbourhood of 0; hence for every $n > 0$, we have $A \not\supseteq n^{-1} V_n$ and there exists a point $x_n$ in $V_n$, such that $n^{-1} x_n \notin A$, that is, $p(x_n) \geq n$. The sequence $(x_n)$ tends to 0, hence is bounded (III, p. 3, corollary); this contradicts the hypothesis on p.