On a current problem sheet I've read the following: Let $G\subseteq\mathbb C$ be a domain and $f:G\to\mathbb C$ holomorphic. Let $(z_n)_n$ be a sequence with a limit point in $G$ and $f(z_n)=0$ for all $n$, then $f(z)\equiv 0$ for all $z\in G$ according to the identity theorem.
My question is this: Is this even true? The sequence $z_n=0$ for all $n\in\mathbb N$ has the limit point $0\in\mathbb C$ and $f(z)=z$ satisfies the condition, however $f$ is not equal to $0$ on $\mathbb C$.
As far as I know the identity theorem needs limit points of sets to be applied. So the statement would be true if we said the following: Let $(z_n)$ be a sequence such that $\{z_n:n\in\mathbb N\}\subseteq G$ has a limit point in $G$. Now if $f(z_n)=0$ for all $n\in\mathbb N$, the identity theorem implies $f\equiv 0$ on $G$.
Am I correct?
Best Answer
The discussion in the comments shows that there is no consensus about the concept of a limit point a of a sequence. This is not a new discussion. Probably the reason for dissenting opinions is that a limit point $p$ of a set $S$ definitely requires the existence of a point $q \in S \setminus \{ p \}$ in any neighborhood of $p$. See for example Limit point of sequence vs limit point of the set containing all point of the sequence, what is diffrernce between limit point of sequence and limit of sequence.
Thus the whole problem is a sort a linguistic problem.
My personal opinion is that RedLantern's interpretation is correct. See also https://www.quora.com/What-are-the-differences-between-limit-and-limit-point. I think the limit of a sequence (if it exists) deserves to be called a limit point of the sequence, and this applies to constant sequences.