Let $K$ be a compact subset of $\mathbb{C}$ and let $E$ be a subset of $\mathbb{C}_\infty=\mathbb{C}\cup\{\infty\}$ (the extended complex plane) which intersects every connected component of $\mathbb{C}_\infty-K$ If $f$ is an analytic function in an open set containing $K$ given $\varepsilon>0$ there is a rational function $R(z)$ whose only poles are in $E$ such that $|f(z)-R(z)|<\varepsilon$ for all $z\in K$