Lef $f:\Bbb{C^{n+1} }\times \Bbb{C} \to \Bbb{C}$ being holomorphic function defined on the polydisc $D'\times D_n$ with $Z(f)$(the vanishing set of $f$) do not have limit point on $D'\times \partial D_n$.
Prove if there is a subset $D'\setminus C \subset D'$ is dense and open, then $Z(f)\cap ((D'\setminus C )\times D_n)$ is dense in $Z(f)$.
The book says it's due to Rouche's theorem, however I don't know to prove it rigourously.
The intuition is clear for each slice $(w',-)$ with $w' \in D'$ fixed, there are only finite intersection points, and union them on nowhere dense set is still nowhere dense. However that's not really a proof.
Focus on the set which is NOT in the zero set. Above a point as you found pick any point not in the zero set. Then pick a neighborhood of that point and use Rouché to show that when you move the $w'$ a bit, there is still no zeros there. You'll get a whole neighborhood that way.