Prove that $f: (a,b)→ℂ$ cannot have infinitely many zeros in $(a,b)$

Question

I have the following nonzero analytic function: $f:ℂ→ℂ$. We will consider only the restriction $f: (a,b)→ℂ$, $a,b∈ℝ$ and $a<b$.

My question is: Prove that $f: (a,b)→ℂ$ cannot have infinitely many zeros in $(a,b)$.

Hint: Use the fact that $f$ is a nonzero entire function and the interval $(a,b)$ have an accumulation point.

Answer 1

As hinted by CutieKrait, if $(a_{n})_{n}$ is a sequence of different zeros of $f$, then it must has a convergent subseqence. so $f$ is constant on a convergent sequence. so it's constant since the interval $(a,b)$ have an accumulation point.