Uniform convergence of random distribution functions

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 $F_{n}(u):=\frac{1}{n} \sum_{i=1}^n 1\{X_i\leq u\}$. We assume that \begin{align*} F_{n}(u)\xrightarrow{d}c(u) Z \end{align*}(in distribution) for $n\rightarrow\infty$. $Z$ is a real valued random variable with continuous distribution and $c: \mathbb{R}\rightarrow \mathbb{R_{>0}}$ is continuous and increasing. Follows now that $F_{n}(u)\xrightarrow{d}c(u) Z$ uniform in $u$ for $n\rightarrow\infty$?