Solution
= Solution
Use the local coordinate $w=1/z$ around infinity. A <pole> at $w=0$ is isolated: for some $\varepsilon>0$, $g(w)=f(1/w)$ is holomorphic at every point $0<|w|<\varepsilon$. Taking $r=1/\varepsilon$, the change of variable maps this punctured disk to $r<|z|<\infty$. Therefore \b[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.