Let $\Gamma=\{z \in \mathbb{C}: z \neq 1,|\operatorname{Re}(z)|+|\operatorname{Im}(z)|=1\}$.

(i) Prove that, for any $\zeta \in \mathbb{C}$ with $|\operatorname{Re}(\zeta)|+|\operatorname{Im}(\zeta)|>1$ and any $\epsilon>0$, there exists a complex polynomial $p$ such that

$$\sup _{z \in \Gamma}\left|p(z)-(z-\zeta)^{-1}\right|<\epsilon$$

(ii) Does there exist a sequence of polynomials $p_{n}$ such that $p_{n}(z) \rightarrow(z-1)^{-1}$ for every $z \in \Gamma ?$ Justify your answer.