Convergence in probability, continuity and uniform convergence in probability

Question

Let $(X_i)_{i\in\mathbb{N}}$ be a strictly stationary sequences of real valued random variables with finite variance. We have the empirical distribution functions $F_{n}(u):=\frac{1}{n} \sum_{i=1}^n 1\{X_i\leq u\}$ and a continuous distribution function $F$. Assume that for all $u\in\mathbb{R}$ \begin{align*} |F_{n}(u)-F(u)|\xrightarrow{p} 0 \end{align*}in probability for $n\rightarrow\infty$.
Holds now that \begin{align*} \sup_{u\in\mathbb{R}} |F_{n}(u)-F(u)|\xrightarrow{p} 0? \end{align*} Thanks!

Answer 1

We can use Corollary 1.4 of paper

Ramon van Handel, The universal Glivenko-Cantelli property, Probab. Th. Rel. Fields 155, 911-934 (2013)

which is available here (especially point 8). The assumption about pointwise convergence allows us to see that for all $u$,
$$F(u)=\mathbb P(\chi_{\{Z_0\leqslant u\}}\mid\mathcal I)\quad\mbox{p.s}.$$