We say a sequence of random variables ${x_n}$ converge in exponential moments to $X$ if $\mathbb E(e^{zX_n})\to \mathbb E(e^{zX})$ uniformly in some neighborhood of $0\in \mathbb C$.
I want to show the convergence in exponential moments implies the convergence in distribution. To this end, it suffices to show that for any $t\in \mathbb R$, we have $\mathbb E(e^{itX_n})\to \mathbb E(e^{itX_0})$. The difficult part is that $U$ may not necessarily contain the whole imaginary axis (we only have the convergence on a neighborhood of $0$). Thanks for help!