In the process of reading a proof on analytic capacity, I'm verifying a detail but am having trouble doing so.
Let $f$ be a bounded analytic function such that $f(\infty)=0$ and $f(z_0)\neq 0$, for some $z_0\neq\infty$. Define $$g(z)=\frac{f(z)-f(z_0)}{z_0-z}$$
The author asserts that $g'(\infty)=f(z_0)$ which is what I'm having difficulty verifying. Here, $g'(\infty)=\lim_{z\to\infty}z(g(z)-g(\infty))$. In particular, I'm not sure what $g(\infty)$ is. Any help is appreciated.
$$g(\infty)=\frac{f(\infty)-f(z_0)}{z_0-\infty} = \frac {-f(z_0)}{-\infty} = 0$$
Then:
$$g'(\infty)=\lim_{z \to \infty} z(g(z)-g(\infty))=\lim_{z \to \infty} zg(z)=\lim_{z \to \infty} \frac {f(z)-f(z_0)}{\frac {z_0}z-1} = \frac {0-f(z_0)}{-1}=f(z_0)$$