We work in $\mathbb{C}$. Consider

$$K=\{z:|z-2| \leqslant 1\} \cup\{z:|z+2| \leqslant 1\}$$

and

$$\Omega=\{z:|z-2|<3 / 2\} \cup\{z:|z+2|<3 / 2\}$$

Show that if $f: \Omega \rightarrow \mathbb{C}$ is analytic, then there is a sequence of polynomials $p_{n}$ such that $p_{n}(z) \rightarrow f(z)$ uniformly on $K$.

Show that there is a sequence of polynomials $P_{n}$ such that $P_{n}(z) \rightarrow 0$ uniformly for $|z-2| \leqslant 1$ and $P_{n}(z) \rightarrow 1$ uniformly for $|z+2| \leqslant 1$.

Give two disjoint non-empty bounded closed sets $K_{1}$ and $K_{2}$ such that there does not exist a sequence of polynomials $Q_{n}$ with $Q_{n}(z) \rightarrow 0$ uniformly on $K_{1}$ and $Q_{n}(z) \rightarrow 1$ uniformly on $K_{2}$. Justify your answer.