$\newcommand{\al}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\si}{\sigma}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\theta}
\newcommand{\om}{\omega}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\F}{\mathcal{F}}
\newcommand{\E}{\operatorname{\mathsf E}} 
\newcommand{\Var}{\operatorname{\mathsf Var}} 
\renewcommand{\P}{\operatorname{\mathsf P}}
\newcommand{\ii}[1]{\operatorname{\mathsf I}\{#1\}} 
\newcommand{\tf}{\widetilde{f}}$ 

It is easy to see (say, using the formulas for the [variance and excess kurtosis for the beta distribution][1]) that, if $Y$ has the beta distribution with parameters $\al,\be>0$, then $\mu_4(Y):=\E(Y-\E Y)^4\ll\al\be/(\al+\be)^5$. So, if $X_1,X_2,\dots$ are iid uniformly distributed on $[0,1]$, then for any real $x>0$ 
\begin{equation}
	\P(\sup_{1 \le i \le n} \De^{(i)} \ge x) \le \sum_1^n \P( \De^{(i)} \ge x) \leq \frac{1}{x^4}\sum_1^n \mu_4(X^{(i)}) \ll \frac{1}{x^4}\sum_1^n \frac{n^2}{n^5} \to0,  
\end{equation}
as desired. 


[1]: https://en.wikipedia.org/wiki/Beta_distribution