= Solution
Let $\mathcal A$ be the <uniform closure> on $K$ of <polynomials>. It is a closed algebra: <uniform convergence> on a <compact set> is preserved under sums and products. For $\lambda\in\Lambda$, put $q(z)=(z-\lambda)^{-1}\in\mathcal A$ and $d=\operatorname{dist}(\lambda,K)>0$. If $|\mu-\lambda|<d/2$, then $\mu\notin K$ and
$$
\frac1{z-\mu}=\sum_{j=0}^\infty(\mu-\lambda)^jq(z)^{j+1}.
$$
The series <converges uniformly>, its ratio having modulus at most one half. Each <partial sum> is in $\mathcal A$, so its limit is too. Hence \b[$\mu\in\Lambda$ in this neighbourhood]. For empty $K$, approximation is vacuous and the assertion is immediate.
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