Consider the sequence $a_n=(1/n)e^{i/(n+1)}$ and the sets $K_n=\{z\in \bar{D}(0,n):1/n\leq \arg z\leq 2\pi\}\cup \{0\}$. Prove that there is a sequence of complex polynomials $P_n:\mathbb{C}\rightarrow \mathbb{C}$ such that $P_n(a_n)=n+1$ and $\|P_n\|_{K_n}<1/2$.
So I was going to use the following consequence of Runge's Theorem:
If $K\subset \mathbb{C}$ is compact and $\mathbb{C}-K$ is connected then any holomorphic map on a neighborhood of $K$ can be approximated uniformly on $K$ by polynomials.
If I use it I can show the statement $\|P_n\|_{K_n}<1/2$ easily. But how do I know or make them so that $P_n(a_n)=n+1$?
Any help is appreciated.
Hint: For each $n$ we can choose disjoint open discs $D_1,D_2$ containing $K_n$ and $\{a_n\}$ respectively. The function that is $0$ on $D_1$ and $n+1$ on $D_2$ is holomorphic on $D_1 \cup D_2.$