Suppose $f$ is meromorphic on $\mathbb{C}_{\infty}$.
Is it true that if $f$ has a pole only at $\infty$ $\iff$ $\operatorname{Res}[f,\infty]=0$ ?
My attempt: If $f$ has a pole only at $\infty$ then $g(z)=f(\frac{1}{z})$ has pole at $0$ of order $n$ say, so $z^ng(z)$ is bounded near $0$ and that means, $z^{-n}f(z)$ is bounded near $\infty$ and hence $f$ is a polynomial. So $\operatorname{Res}[f,\infty]=0$.
Am I right?