This is a technical lemma that I saw which I don't know how to prove. The simplified context is the following:
Let U connected open set in $\mathbb{C}^n$, $\pi: S\to U$ be a finite k-sheeted covering (i.e. surjective, local biholomorphism), $S$ is the covering space. Now find a countable dense set $\{x_n\}$ of $U$. The goal now is to find a linear function $l$ such that $l$ separate the $k$ points in each fibre $\pi^{-1}(x_n)$, for all $x_n$.(Note that $l$ doesn't necessarlly have to separate points in different fibres)
What I know so far, from linear algebra, is that if you give me any finite number of points, I can find a linear function in $\mathbb{C}^n$ to separate them, so the question really is, how to find a uniform $l$ such that the above holds.
Just in case you're interested in the context, the question comes from Page 51, introduction to the theory of Analytic space by Narasimhan. Thanks in advance for your help!