I have a small doubt related to the identity theorem
Let $f(z) $ be an entire function and there exist some nonconstant sequence $z_n$ such that\begin{equation} z_n = \begin{cases} 1/n & \text{if }n = 2m, m \in \mathbb{N}\\ n & \text{if } n = 2m-1, m \in \mathbb{N} \end{cases} \end{equation} such that $f(z_n)=0 $.
Means we say the analytic function is constant if the limit point of the sequence of zeros lies inside the domain of analyticity of the function. My doubt is whether is it necessary for entire functions to be constant (Using identity theorem) the all the subsequences of the sequence of zeros are convergent only Is it enough to have one subsequence whose limit belongs to the domain of analyticity?
Question: Is $f(z) $ a constant function?
My guess is "yes" the function is identically zero in such condition also and it is because "when some analytic function in domain D have zero that means there is a small disc (say$\Delta$) around that zero such that $f(z)=0 \forall z \in \Delta$
I'm on the right track,
Any comment is well appreciated.