Suppose $f$ is a complex analytic in a neighbourhood of $0$ such that $\forall k\in \mathbb{N}(\exists n_0 (\forall n>n_0( |f(a_n)|\leq {|a_{n}|}^k)))$ where $a_n$ is a non-zero sequence converging to $0$, can we say that $f$ is $0$?
I tried for specific examples of $a_n$, it feels like true when this statement "function matches at all points if laurent series matches at infinitely many points with an accumulation point" is true.