[CMI PG2010, Part B] 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 $n$.
Since $f$ is analytic on $U$, we can have a power series expansion of $f$, say $f(z)= \displaystyle\sum_{n=0}^{\infty}{r_n}{z^n}$. Repalcing $z$ by $a_j\forall j$, we get $f(a_j)= \displaystyle\sum_{n=0}^{\infty}{r_n}{{a_j}^n}=b_j$. Hence we have, $r_0 = f(0) = 0$. Now I am stuck here. Please give me hint how to proceed.
Thanks in advance!!
Hint: If $f$ is nonzero, then $f(z)=z^kg(z)$ for some $k\ge 0$ and some analytic function $g$ in $U$ such that $g(0)\ne 0$. Therefore, $$\lim_{z\to 0}\frac{f(z)}{z^K}=\lim_{z\to 0}g(z)=g(0)\ne 0.$$