Past exam of the Probability, statistics and options II course of the University of Oxford / 2007 / D / 4 / ii / Solution

For fixed $x>0$, $f_n(x)\to e^{-x}\sin x$; for $x\le 0$, the limit is zero. On $0<x<n$, the supplied inequality gives $(1-x/n{)}^n\le e^{-x}$, soThis is the given integrable dominating function. By the dominated convergence theorem,