(a) Let $K$ be a closed subset of the unit disc in $\mathbb{C}$. Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a rational function with all its poles of modulus strictly greater than 1 . Explain why $f$ can be approximated uniformly on $K$ by polynomials.

[Standard results from complex analysis may be assumed.]

(b) With $K$ as above, define $\Lambda$ to be the set of all $\lambda \in \mathbb{C} \backslash K$ such that the function $(z-\lambda)^{-1}$ can be uniformly approximated on $K$ by polynomials. If $\lambda \in \Lambda$, prove that there is some $\delta>0$ such that $\mu \in \Lambda$ whenever $|\lambda-\mu|<\delta$.