# Lévy and Lévy Prokhorov metric [duplicate]

I was reading "On choosing and bounding probability metrics" (https://www.math.hmc.edu/~su/papers.dir/metrics.pdf) and encountered the Lévy distance,defined by

$d_L(\mu,\nu):=\inf\lbrace{}\varepsilon>0\,:\, \mu((-\infty,x-\varepsilon])\leq\nu((-\infty,x])\leq\mu((-\infty,x+\varepsilon]), \forall x\in\mathbb{R}\rbrace$

as a special case (on $\mathbb{R}$) of the Lévy-Prokhorov distance, defined by

$d_P(\mu,\nu):=\inf\lbrace\varepsilon>0\,:\,\mu(B)\leq\nu(B^{\varepsilon})+\varepsilon,\forall \text{ measurable }B\subset\mathcal{X}\rbrace$

In the transcript it said that $d_P$ metrized weak convergence. It is clear that $d_L\leq{}d_P.$

Additionally it mentioned that $d_L$ metrizes weak convergence too, which I don't quite understand since it is only bounded from above by Lévy-Prokhorov. For it to metrize weak convergence it should have to be equivalent to $d_P$, which I don't believe it is. Or maybe I'm mistaken. Can anyone explain?

Thanks a lot.