Theorem 3 (Runge)

Let K be a compact subset of $\mathbf{C}$, and Q the set of germs of rational functions holomorphic in a neighbourhood of K. Then Q is dense in $\mathscr{O}(K)$.

By the definition of the topology on $\mathscr{O}(K)$, it is enough to prove that for every open neighbourhood U of K, and every compact subset L of U, every function $f\in \mathscr{O}(U)$ is the limit of rational functions continuous on L. One may suppose that L is a compact neighbourhood of K.

Let $Q'$ be the set of restrictions to L of rational functions on $\mathbf{C}$ which are continuous on L, and let C be the closure of $Q'$ in $\mathscr{C}(L)$. It is an algebra with no radical.

Let $z\in C$ be the identical mapping of L. Then C is the full closed subalgebra of $\mathscr{C}(L)$ generated by $z$ (lemma 2 of I, p. 6). Hence Sp$_C(z) =$ Sp$_{\mathscr{C}(L)}(z) = L$. One can therefore form the element $c= \Theta_z(f)$ of C. Since C has no radical, application of cor. 1 of I, p. 66 to the characters $g\mapsto g(w)$ of C, for every $w\in L$, shows that $c$ coincides with the restriction of $f$ to L. By definition of C, this proves that $f|L$ is the uniform limit on L of elements of $Q'$, and this completes the proof of the theorem.