Show that if \(f_n\to f\) locally uniformly and each \(f_n\) is holomorphic then \(f\) is holomorphic.
(Variations)
If \(f_n\to f\) uniformly on all of \(\Omega\), the hypothesis holds a fortiori, since uniform convergence on \(\Omega\) is uniform convergence on each compact subset; the proof above applies unchanged.
Solution
Let \(\Omega\) be the open set on which the \(f_n\) are holomorphic.
1. \(f\) is continuous on \(\Omega\).
Each point of \(\Omega\) has a compact neighbourhood in \(\Omega\), on which \(f\) is a uniform limit of continuous functions.
2. \(\int_{\partial T}f=0\) for every closed triangle \(T\subset\Omega\).
By Goursat’s theorem \(\int_{\partial T}f_n=0\). The boundary \(\partial T\) is compact, so \(f_n\to f\) uniformly on it, and \(\int_{\partial T}f=\lim_n\int_{\partial T}f_n=0\).
3. Q.E.D.
Every point of \(\Omega\) has a disc in \(\Omega\) around it, and on that disc steps 1 and 2 are the hypotheses of Morera’s theorem [@SS03, Chapter 2, Theorem 5.2]. So \(f\) is holomorphic near every point of \(\Omega\).