In the proof of Cartan Theorem B on Hörmander's SCV book (p.199 in third edition), it seems that the following claim is made without justification:
Let $\Omega$ be a Stein manifold of dimension $n$. Then there exist arbitrarily fine open coverings $\{U_i\}_{i\in I}$ of $\Omega$ such that each $U_i$ is a Stein manifold, and more than $N$ ($N>2n$ given) sets in $\{U_i\}_{i\in I}$ always have an empty intersection.
In the book, the author first chooses $\{U_i\}_{i\in I}$ with each $U_i$ Stein. Then, to conclude that certain sheaf cohomology groups vanish, the author writes:
But if the covering is chosen so that more than $N$ sets $U_i$ always have an empty intersection, which is possible if $N>2n$, it is obvious that ...
He then proceeds without explaining why that is possible.
Why is this true? The best I can do is choose $\{U_i\}_i$ to be paracompact, so that on each compact set such an $N$ exists. However, this argument does not give a uniform $N$, let alone a precise bound related to the dimension. Am I missing something?
Any help is appreciated.