= Solution
The polynomial form of the <Runge approximation theorem> says: if $K\subset\mathbb C$ is <compact>, $\mathbb C\setminus K$ is connected, and $f$ is <holomorphic> on a neighborhood of $K$, then for every $\varepsilon>0$ there is a <polynomial> $P$ with $\sup_K|f-P|<\varepsilon$. Choosing errors tending to zero gives a uniformly convergent sequence of <polynomials>. Equivalently, on an open domain with connected complement in the Riemann sphere, every <holomorphic function> is the compact-uniform limit of <polynomials>. The connected-complement hypothesis excludes enclosed holes containing poles of the function being approximated.
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