Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 201 / 3 / d / Solution

Yes. If a subsequence converges weakly to $\nu$, bounded continuity of $e^{iuy}$ givesThe uniqueness theorem for characteristic functions makes $\nu$ unique. Tightness implies that every subsequence has a further weakly convergent subsequence, and every such limit is $\nu$. This subsequence criterion proves that the entire sequence converges weakly to $\nu$.