Let $\{a_n\}$ and $\{b_n\}$ be sequences of complex numbers such that each $a_n$ is non-zero, $\lim_{n \rightarrow \infty} a_n = \lim_{n \rightarrow \infty} b_n = 0$, and such that for every natural number $k$,
$$\lim_{n \rightarrow \infty} \frac{b_n}{a_n^k} = 0$$
Suppose $f$ is an analytic function on a connected open subset $U$ of $\mathbb{C}$ which contains $0$ and all the $a_n$. Show that if $f(a_n) = b_n$ for every natural number $n$, then $b_n = 0$ for every natural number $n$.