Given a series of polynomials $p_{n}$ and a series of non-intersecting balls $B_{n} \subset \mathbb{C}$ show that there exists some function $f \in \mathcal{O}(\mathbb{C})$ such that $lim_{n \rightarrow \infty} sup_{z \in B_{n}} |f(z)-p_{n}(z)|=0$.
Normally approximation is the other way around, so I'm having a little trouble with this.
The claimed result is false:
Let $B_n$ be an open disk with center $\frac 1n$ and of radius $r_n$ sufficiently small to ensure that the disks do not contain $0$ and are disjoint, and define $p_n(z)=n$, the constant polynomial with valiue $n$.
If $f\in \mathcal O(\mathbb C)$ existed we would have on one hand $f(0)=\lim_{n \rightarrow \infty}f(\frac 1n)$.
On the other hand we would have $lim_{n \rightarrow \infty} \vert f(\frac 1n) -n)\vert =0$ [because of the hypothesis $lim_{n \rightarrow \infty} sup_{z \in B_{n}} |f(z)-p_{n}(z)|=lim_{n \rightarrow \infty} sup_{z \in B_{n}} |f(z)-n|=0$] and thus in particular $lim_{n \rightarrow \infty} \vert f(\frac 1n) )\vert=\infty$.
Since the two hands cannot agree , the function $f$ does not exist !