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Apr 25, 2012 at 14:48 comment added Per Alexandersson Yes, I do mean subsequence, and your comment do imply that I need extra conditions on P and Q. What suitable conditions I need is what I am asking for. This notion of limit (with subsequences) of set have been used in various places, to I'd like to keep that.
Apr 25, 2012 at 13:53 comment added Andreas Blass Do you really mean "subsequence" in the definition of $C(y^*)$? If you do, then it seems that you could have $x\in C(y)$ because of zeros of the $P_n$'s for even $n$, and you could have $y\in C(x)$ because of zeros of the $Q_n$'s for odd $n$, in which case you seem to have no information relevant to $P_n=Q_n=0$ (with the same subscript $n$ on both $P$ and $Q$).
Apr 25, 2012 at 12:38 history asked Per Alexandersson CC BY-SA 3.0