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
By Morera’s theorem, it suffices to show \(\int_T f = 0\) for all triangles \(T \subseteq \DD\). Noting that \(T\) is closed and bounded and thus compact, \(f_n\to g\) uniformly on \(T\). Since the \(f_n\) are holomorphic, \(\int_T f_n = 0\) for all \(n\), and thus \[ \int_T g = \int_T \lim f_n = \lim_n \int_T f_n = \lim_n 0 = 0 ,\] where moving the limit through the integral is justified by uniform convergence.