Choice bookkeeping for Runge and Mittag-Leffler

The proofs on this page separate two kinds of choice.

First, the compact-set Runge theorem is finitary once the pole representatives are named: the enclosing cycle is a finite polygonal chain, the Riemann sums use finite sampling, and pole pushing moves finitely many poles along finitely many disc chains.

Second, the domain versions need an exhaustion and one representative in each component of $\widehat{\mathbb{C}}\setminus \mathrm{\Omega }$. The exhaustion can be chosen canonically, but the complementary representatives are genuine extra data unless the domain already comes with a distinguished choice, for example when the complement is connected and $\{\mathrm{\infty }\}$ is the pole set. The local proofs therefore stay within ordinary finitary reasoning, while the global statements record exactly where the component-by-component bookkeeping enters.