Let $\left(X_{n}: n \in \mathbb{N}\right)$ be a sequence of independent normal random variables having mean 0 and variance 1 . Set $S_{n}=X_{1}+\ldots+X_{n}$ and $U_{n}=S_{n}-\left\lfloor S_{n}\right\rfloor$. Thus $U_{n}$ is the fractional part of $S_{n}$. Show that $U_{n}$ converges to $U$ in distribution, as $n \rightarrow \infty$ where $U$ is uniformly distributed on $[0,1]$.