Paracompactness of the projectified bundle over a paracompact space Consider a complex rank $n$ Vector Bundle $V \rightarrow X$. It is a standard Argument in the construction of Chern classes to consider the orthogonal complement to the tautological line bundle carried by $P(V)$, where $P(V)$ is the projectified bundle over $X$ with fiber $\mathbb C P^{n-1}$, basically $V \setminus X$ with $\mathbb C$-multiplication factored out.
I have found numerous references to the fact that $P(V)$ is paracompact, provided that $X$ is, but no rigorous proof. The idea must somehow be to use that products with one factor paracompact and the other compact are paracompact, which would take care of the case $V \cong  X \times \mathbb C^n$. As to other cases, say, $X= U_1 \cup U_2$ and $V$ is trivial over the $U_i$, my problem would be that the $U_i$ need not be paracompact, as they are open in $X$. Perhaps there is a way around this by looking at gluing constructions to recover $V$ as $(U_1 \times \mathbb C^n) \dot\cup (U_2 \times \mathbb C^n)/ \sim$, but I am stuck. Any pointers?
 A: I think I got it. The idea is to take an open cover $\{U_\alpha\}_{\alpha \in J}$ and first prove a "Shrinking" lemma for paracompact spaces. Even though $X$ is normal, given its paracompactness, the standard Shrinking lemma (for a proof see Munkres) needs a finite cover (I guess the proof goes through for a countably infinite one as well). But we can find a locally finite refinement with same index set, $\{V_\alpha\}_{\alpha \in J}$ such that it still is a cover and $V_\alpha \subset \bar{V}_\alpha\subset U_\alpha$. Now if $V \rightarrow X$ is trivial over each $U_\alpha$, then certainly over each $\bar V_\alpha$, too.
But closed subsets of paracompact spaces are paracompact, and since $P(V)|_{\bar V_\alpha} \cong \bar V_\alpha \times \mathbb CP^{n-1}$, the disjoint union $\amalg_\alpha (\bar V_\alpha \times \mathbb CP^{n-1})$ is paracompact, for products consisting of a paracompact and a compact space are paracompact, and disjoint unions of paracompact spaces are paracompact. Now the projections induced from the gluing construction from which we recover $P(V) \rightarrow X$ are closed, hence by Michael's theorem $P(V) \cong \amalg_\alpha (\bar V_\alpha \times \mathbb CP^{n-1})/\sim$ is paracompact.
