I was given this proof:
Let $w(z)=1/z$, so $w$ maps origin to inifinity and infinity to origin.
Consider $f(z) = z$. It has no singularities in finite $z$-plane.
So $f(w) = 1/w$ has a pole at the origin of $w$-plane, which is at infinity in the $z$-plane.
$f(z)=1/z$ has a pole at origin.
So $f(w) = w$ is analytic at $w=0$.
Therefore $1/z$ is analytic at infinity.
I do not understand this proof at all!