Show that if $f$ is a meromorphic function on $\mathbb{C}$ such that $\mathbb{C}_\infty-f(\mathbb{C})$ has at least three points then f is constant.
I constructed a new function $g$ such that $g(z) =f(z)$ if $z$ is not a pole of $f$ and $g(z) = \infty$ if $f$ has a pole at $z$. Then clearly $g$ is entire function and misses at least two points.So by the little Picard Theorem.$g$ must be a constant,so is $f$. Am I correct ?
Thanks in advance