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 $\{#b^{-1}(y):y \in Y\}$ 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.