Let $\Omega$ be a region in $\mathbb{C}$. Let $\{a_n\}$ be a sequence of nonzero elements in $\Omega$ such that $a_n \rightarrow 0$ as $n \rightarrow \infty$. Let $\{b_n\}$ be a sequence of complex numbers such that $\lim_{n \rightarrow \infty} \frac{b_n}{a_n^k} = 0$ for every nonnegative integer $k$. Suppose that $f : \Omega \rightarrow \mathbb{C}$ is an entire function such that $f(a_n) = b_n$ for all $n$. Show that $b_n = 0$ for every $n$.