Start from a countable dense subset $S$ of the unit ball, and take a sequence $Z_\nu$ such that for all $s\in S$, one has $\nu Z_\nu=s$ infinitely often. Then, any holomorphic function in $\mathbb{C}^n$ that vanishes along the sequence $Z_\nu$ has in particular a subsequence of zeros in accumulating to the origin that belong to the complex line generated by any $\mathbb{C}s$, hence s\in S$. Therefore it vanishes identically on that complex line, by the principle of isolated zeros in one variable, and therefore . Since the union of these lines is dense in $\mathbb{C}^n$, the function is identically zero by densitycontinuity.
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Start from a countable dense subset $S$ of the unit ball, and take a sequence $Z_\nu$ such that for all $s\in S$, one has $\nu Z_\nu=s$ infinitely often. Then, any holomorphic function in $\mathbb{C}^n$ that vanishes along $Z_\nu$ has in particular a subsequence of zeros in the complex line $\mathbb{C}s$, hence vanishes identically on that complex line by the principle of isolated zeros in one variable, and therefore is identically zero by density. |
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