= Elementary polynomial approximation on a square
Let $S$ be a closed square and let $f$ be holomorphic on a neighbourhood of a slightly larger square. The <Cauchy integral formula> on the larger boundary is uniformly approximated on $S$ by Riemann sums of resolvents whose poles lie outside $S$. Far-away resolvents have convergent geometric-series expansions in $z$, and <resolvent propagation across a complementary component> moves this polynomial approximability to every pole outside the square. Approximating the finitely many resolvents in each Riemann sum by polynomials proves polynomial approximation on $S$ without invoking the general <Runge theorem>.
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