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:
converges when $|z-z_0|<R$. In our case $R$ is infinity so the only way for the sum to converge is if $a_j=0$. Does this make sense?