First give some notation and definitions:
Let $U_1,\dots,U_n$ be iid $Uniform[0,1]$ distributied and let $0=U_{0,n}\leq U_{1,n}\leq\cdots\leq U_{n,n}\leq U_{n+1,n}=1$ be their order statistics.
For $t\in[0,1]$, let $G_n(t)$ be the sample cdf and let $Q_n(t)$ be the sample quantile function, which is defined as $$Q_n(t)=U_{k,n} \mbox{ if } \frac{k-1}{n}<t\leq \frac{k}{n}$$
and $Q_n(0) = U_{1,n}.$ The uniform empirical process is defined as $a_n(t)=\sqrt{n}(G_n(t)-t)$ and the uniform quantile process is defined as $u_n(t) = \sqrt{n}(t-Q_n(t))$.
My question comes from Equiation (11) of the 1986 paper Approximations of weighted empirical and quantile processes by Csorgoo and Horvath, which is a 6-page-long paper mainly to give a shorter proof of a Theorem in one of their previous papers.
Equiation (11) there says that: with probability one for each n, $$\sup_{0\leq t\leq 1}|u_n(G_n(t))-a_n(t)|\leq n^{-1/2}.$$ I think this Equation may not be right and here's what I've tried: It is easy to show that \begin{align} \sup_{0\leq t\leq 1}|u_n(G_n(t))-a_n(t)|& = \sup_{0\leq t\leq 1}\sqrt{n}|t-Q_n(G_n(t))|\\ & = \sup_{k=1,2,\dots,n+1}\sqrt{n}D_{k,n} \end{align}
where $D_{k,n}=U_{k,n}-U_{k-1,n}$ is the spacing between 2 order statistics.
From this post On the largest and smallest spacings for the uniform distribution, it can be shown that $(n/\ln n) \sup_{k=1,2,\dots,n+1}D_{k,n}\stackrel{p}{\to}1$, which is a direct violation of the claim in Equation (11).
In fact, I think the proof in the original paper will go through if $\sqrt{n}\sup_{0\leq t\leq 1}|u_n(G_n(t))-a_n(t)| = O_p(1)$, but if the above result ($(n/\ln n) \sup_{k=1,2,\dots,n+1}D_{k,n}\stackrel{p}{\to}1$) is correct, even this is not true.
I don't know if there's any mistake in my derivation, or the original proof is actually not right. any comment is appreciated!!