An application of Runge's Theorem to approximate analytic functions by polynomials

Question

Apply the following form of Runge's Theorem: if $X\subset \mathbb{C}$ is an open subset,and if $\mathbb{C}\setminus X$ is connected,then $\mathbb{C}[z]$ is dense in $\mathcal{O}(X)$ in the topology of Frechet space. It follows that,for each $j\in \mathbb{N}$,there exists a $P_j\in \mathbb{C}[z]$ such that \begin{equation} \|P_j\|_{A(j)}<1/j \end{equation} and \begin{equation} \|1-P_j\|_{B(j)}\le1/j \end{equation} where $A(j)=\{z\in\mathbb{C}:|z|<j,\text{Im}\,z>1/j\}$ and $B(j)=\{z\in\mathbb{C}:|z|<j,\text{Im}\,z<1/2j\}$. I don't know how to construct $P_j$ to satisfy the above two inequalities simultaneously.Can you help me?Thanks in advance!

Answer 1

Choose $X$ as a set (with two components) such that $A(j) \cup B(j)$ is relatively compact in $X$ and such that the complement of $X$ is connected. The function $f$ which is $0$ on the component containing $A(j)$ and $1$ on the component containing $B(j)$ is holomorphic on $X$, so by Runge your $P_j$ exists.