My question is the following:

Let $b:X \rightarrow Y$ be a proper holomorphic mapping between two Stein manifolds $X$ and $Y$. For obvious reasons, $b^{-1}(y)$ are finite subsets of $X$. Is the set $\lbrace \# b^{-1}(y)\, \colon\, y \in Y\rbrace $ bounded?

Here is the background: I'm trying to understand the equisingular stratification lemma in the paper "Eliashberg and Gromov: Embeeding of Stein manifold of dimension $n$ into the affine space of dimension $\frac{3}{2}n+1$, Annals of mathematics 1992".The anwser to the above question is yes according to their lemma.And if the anwser is yes,I can prove the existence of equisingular stratification.

A possible way to solve the question is: Denote by $A_i$ the subset of $Y$ consists of points $y \in Y$ with $\#b^{-1}(y) \geq i$, then they are analytic subsets of $Y$. If the number of irreducible components of $A_i$ are finite,then the anwser is yes. The problem is that $Y$ is not a compact manifold and $A_i$ may have infinite many components.