Let $(X_n)_n$ be a sequence of real random variable converging in distribution to $X.$
Prove that $\varphi_{X_n}$ converges uniformly to $\varphi_X$ on every compact $K \subset \mathbb{R},$ by considering an integration by parts for $\int e^{ixy}dP_{X_n}(x).$
Can't see how to integrate by parts. Any hints?