Exercise 30, § 4

In a regular space X, let $\mathscr{F}$ be a locally finite set of sets, each of which is a paracompact subspace of X. Now prove that the subspace union of the sets of $\mathscr{F}$ is paracompact (use Exercise 25 c) of IX, p. 107) (cf. IX, p. 113, Exercise 14).