Solution
= Solution
The <polynomial Runge theorem> states that, for a compact set $K\subset\mathbb C$ with connected complement and a function $f$ <holomorphic> on a neighborhood of $K$, for every $\varepsilon>0$ there is a <polynomial> $P$ such that
$$
\boxed{\sup_{z\in K}|f(z)-P(z)|<\varepsilon.}
$$
The connected-complement hypothesis permits <polynomial> approximation rather than approximation by rational functions with poles in bounded complementary components.