Past exam of the mathematics course of the University of Cambridge / 2003 / ib / Paper 4 / 4E / b / Solution

The uniform limit theorem first gives continuity of $f$ on $D$. For any triangle $\Gamma$ whose closed interior is in $D$, the Cauchy integral theorem gives ${\int }_{\Gamma }{f}_n(z)dz=0$. Moreover,Hence ${\int }_{\Gamma }f=0$, and Morera's theorem proves that $f$ is analytic on $D$. In fact, the proof only needs locally uniform convergence on compact subsets, since each triangle is compact; the stated uniform convergence on the entire domain is stronger.