Suppose \(\theset{f_n}_{n\in \NN}\) is a sequence of analytic functions on \(\DD \definedas \theset{z\in \CC \suchthat \abs{z} < 1}\).
Show that if \(f_n\to g\) for some \(g: \DD \to \CC\) uniformly on every compact \(K\subset \DD\), then \(g\) is analytic on \(\DD\).
Solution
Local uniform convergence implies that \(g\) is continuous. Let \(T\) be any triangle whose closure lies in \(\mathbb D\). Since \(f_n\to g\) uniformly on \(\partial T\), \[ \int_{\partial T}g(z)\,dz =\lim_{n\to\infty}\int_{\partial T}f_n(z)\,dz. \] Each integral on the right is zero by Cauchy’s theorem because \(f_n\) is holomorphic. Hence \[ \int_{\partial T}g(z)\,dz=0 \] for every such triangle. Morera’s theorem therefore implies that \(g\) is holomorphic on \(\mathbb D\).