complex analysis - Convergence of Laurent Series.

Let $f(z)$ be analytic for $|z|>r$ and let it be bounded $|f(z)|\leq M, M>0$ wherever it is analytic. Show that the coefficients of the Laurent Series of $f(z)$ are $0$ for $j\geq 1$.

I have found two approaches to solve this. I'm not sure about this one:

The positive part of the Laurent Series:

$$\sum_{j=0}^\infty a_j(z-z_0)^j$$

converges when $|z-z_0|