real analysis - Proof for power series

I used half my Sunday for trying to proof the following, but I couldn't find the answer. Can you help me?

Let $f(x)=\sum_{k=0}^{\infty}a_k(x-x_0)^k$ a power series with radius of convergence $\ge 0$. Show: $f(x_n)=0$ for a sequence of points $\{x_n\}$ with $x_n \rightarrow x_0, x_n \neq x_0 \implies a_k = 0$ $\forall k \in \mathbb{N}$.

Hint: Define for $j \in \mathbb{N}$
$$f_{(j)}(x):= \sum_{k=0}^{\infty}a_{j+k}(x-x_0)^k$$ and use induction to prove for all $j \in \mathbb{N}$:
$f_{(jn)}(x_n)=0$ for all $n \in \mathbb{N}$ and $a_j=0$.