 Solution

Use the local coordinate $w=1/z$ around infinity. A pole at $w=0$ is isolated: for some $\epsilon >0$, $g(w)=f(1/w)$ is holomorphic at every point $0<|w|<\epsilon$. Taking $r=1/\epsilon$, the change of variable maps this punctured disk to $r<|z|<\infty$. Therefore there are no finite poles outside that disk. This is a property of the local meromorphic chart at infinity, not an assertion that infinity itself is regular.