# Prove existence of inverse of analytic function in some neighborhood

Suppose $$f(z)$$ is analytic at $$z_{0}$$ with $$f^{\prime}\left(z_{0}\right) \neq 0 .$$ Show that there exists an analytic function $$g(z)$$ such that $$f(g(z))=z$$ in some neighborhood of $$z_{0}$$.

This is also known as inverse function theorem. I am looking for the analytical proof that does not use neither Jacobian nor inverse function theorem for $$R^2$$.

I have tried to use Open Mapping Theorem, but couldn't prove that mapping is bijective. Any help or tips?