Let X be a paracompact Hausdorff space. $M=\bigcup_{i=1}^{\infty}F_i$, $F_i$ is closed. Prove that M is paracompact as a subspace of X.
Idea: In Munkres' book topology, there are two theorems:
(1) Every paracompact Hausdorff space is normal.
(2) Every closed subspace of a paracompact space is paracompact.
In my problem, M is a countable union of closed sets. By (2), we know that M is a countable union of closed paracompact subspaces. For any open covering $\mathscr{A}$ of M, we need to show $\mathscr{A}$ has a refinement that is an open covering of M and locally finite. I want to find an open covering $\mathscr{B}$ of X satisfying $\mathscr{A}\subset\mathscr{B}$ and $\forall V\in\mathscr{B}\backslash\mathscr{A}, V\bigcap A=\phi$. If I can find such $\mathscr{B}$, I can use the condition that X is paracompact. But I can't find such one.
Any ideas?