 Locally uniform convergence of holomorphic functions

A locally uniform convergence limit of holomorphic functions is holomorphic, and its derivatives converge locally uniformly as well. The limit is continuous. On each small triangle contained in the domain, uniform convergence allows passage to the limit in its contour integral; the integrals vanish by the Cauchy integral theorem. The Morera theorem therefore makes the limit holomorphic. Applying the Cauchy integral formula on circles in a slightly larger compact subset proves locally uniform convergence of every fixed-order derivative.