Quantile coupling for convergence in distribution
= Quantile coupling for convergence in distribution
If $F_n\to F$ at every continuity point of $F$, then the generalized inverses satisfy
$$
F_n^{-1}(u)\to F^{-1}(u)
$$
at every continuity point of $F^{-1}$. Since a monotone function has only countably many discontinuities, evaluating all quantiles at one uniform random variable gives an almost-sure coupling of the corresponding convergence in distribution.