Entire function with arbitrary zeroes $(a_n)$ but $|a_n| \to a \neq \infty$

Question

I've been studying the Weierstrass factorisation theorem and in the proofs I see they assume that the zeroes of an an entire function always tend to infinity.

Is it possible to define an entire function with arbitrary zeroes $(a_n)$ but $|a_n| \to a \neq \infty$ or just have $\lim_{n \to \infty}a_n$ not exist at all? for example, could there be an entire function that has $e^{ni}$ as zeroes for all $n \in \mathbb{N}$?

Answer 1

If $f\colon\Bbb C\longrightarrow\Bbb C$ is an entire function which is not the null function and $(a_n)_{n\in\Bbb N}$ is a sequence of distinct zeros of $f$, then we always have $\lim_{n\to\infty}|a_n|=\infty$. Otherwise, the sequence $(a_n)_{n\in\Bbb N}$ would have a subsequence $\left(a_{n_k}\right)_{k\in\Bbb N}$ that would converge to some $a\in\Bbb C$ and then, by the identity theorem, $f$ would be the null function.

Answer 2

$(a_n)$ is bounded, so it has a limit point. It follows that $f$ is identically $0$ unless $\{a_n: n \geq 1\}$ is a finite set in which case there is a polynomial solution.