Past exam of the mathematics course of the University of Cambridge / 2007 / ib / Paper 4 / 3H / ii / Solution

For each fixed $x$, $x/n\to 0$, so the pointwise limit is $1$. Moreover $1-\cos u=2{\sin }^2(u/2)\le u^2/2$ givesThus the limit is $1$, with uniform convergence on $(0,1)$.