MHB A whole function approximating polynomials

[LWRS]

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 other way around so I'm not sure what to do here.

 

[Jhoneine]

Let g(z) denote the function we want to construct. We can use a technique known as the Weierstrass Approximation Theorem to construct g(z). This theorem states that if V_{n} is compact and p_{n} is continuous, then there exists a sequence of polynomials f_{n} such that lim_{n \rightarrow \infty} sup_{z \in V_{n}} |f_{n}(z)-p_{n}(z)|=0.Now, we can define g(z) as the uniform limit of the sequence of polynomials {f_{n}(z)}, i.e., g(z)=lim_{n \rightarrow \infty} f_{n}(z). Since f_{n}(z) converges uniformly to p_{n}(z), it follows that lim_{n \rightarrow \infty} sup_{z \in V_{n}} |g(z)-p_{n}(z)|=0. Hence, we have constructed a function g(z) which satisfies the desired condition.