SCHWEITZER 2014 Problem 11

Let $U$ be a random variable that is uniformly distributed on the interval $[0,1]$ , and let $$S_n= 2\sum_{k=1}^n \sin(2kU\pi).$$ Show that, as $n\to \infty$ , the limit distribution of $S_n$ is the Cauchy distribution with density function $f(x)=\frac1{\pi(1+x^2)}$ .