  We consider a sequence of identically independently distributed random
samples from an absolutely continuous probability measure in one dimension with
unbounded density. We establish a new rate of convergence of the
$\infty-$Wasserstein distance between the empirical measure of the samples and
the true distribution, which extends the previous convergence result by
Trilllos and Slep\v{c}ev to the case that the true distribution has an
unbounded density.
