Consider a sequence of iid standard uniform random variables $U_1,\ldots, U_n$ and their order statistics $U_{1,n}\leq \ldots \leq U_{n,n}$.
I am led to believe that \begin{equation*} 1- U_{n,n} = O\mathopen{}\left(\frac{\log\log n}n\right), \quad \text{a.s. for }n\to \infty, \end{equation*} where $O(\cdot)$ is the notation for a big-$O$ function, i.e. supposedly (as I understand it), \begin{equation*} \limsup_{n\to\infty}\frac{n(1-U_{n,n})}{\log\log n} < \infty \quad \text{a.s.} \end{equation*}
How do I go about to prove this?
Backstory:
In an article1 I am reading they mention that the above result holds by by Kiefer (1972), however, I cannot find that citation in their bibliography and searching for it (attempt 1, attempt 2, attempt 3), I at maximum find Amazon/Google books results that also refer to the citation but I have not had any luck at having page reviews of their bibliographies. As I cannot find a reference to see the proof and because I would like to use the result, I would like to prove it.
1. Paul Deheuvels, David M. Mason, and Galen R. Shorack. Some results on the influence of extremes on the bootstrap. Annales de l’I.H.P. Probabilités et statistiques, 29(1):83–103, 1993. http://eudml.org/doc/77452