Given a series of polynomials $$p_{n}$$ and a series of open, non-intersecting sets $$V_{n} \subset \mathbb{C}$$ show that there exists a function $$g\in \mathcal{O}(\mathbb{C})$$ such that $$lim_{n \rightarrow \infty} sup_{z \in V_{n}} |g(z)-p_{n}(z)|=0$$.
Normally the approximation goes the...