Prove the uniform limit theorem for holomorphic functions: if \(f_n\to f\) locally uniformly and each \(f_n\) is holomorphic then \(f\) is holomorphic.
Solution
This is S&S Theorem 5.2. Statement: if \(f_n\to f\) uniformly locally uniformly on \(\Omega\) then \(f\) is holomorphic on \(\Omega\).
- Let \(D \subset \Omega\) with \(\bar\DD \subset \Omega\) and \(\Delta \subset D\) be a triangle.
- Apply Goursat: \(\int_\Delta f_n = 0\).
- \(f_n\to f\) uniformly on \(\Delta\) since it is closed and bounded and thus compact by Heine-Borel, so \(f\) is continuous and \[ \lim_n \int_\Delta f_n = \int_\Delta \lim_n f_n \da \int_\Delta f .\]
- Apply Morera’s theorem: \(\int_\Delta f\) vanishes on every triangle in \(\Omega\), so \(f\) is holomorphic on \(\Omega\).