Convergence of uniform distributed order statistic

Question

I am working on this problem: Suppose $X_1, X_2...$ i.i.d~ $U(0,1)$, prove that $n^{-1}\log(1-X^{n-1}_{(n)})$ converges to $0$ in probability. I can derive the distribution function of $X^{n-1}_{(n)}$, and get the distribution of $n^{-1}\log\left(1-X^{n-1}_{(n)}\right)$, which equals $1-(1-\exp(nt))^{\frac{n}{n-1}}$, but I can not work out the limitation of this. What should I do next or are there any other thought to solve this problem?

Answer 1

First observe that the random variable $Y_n:=n^{-1}\log(1-X^{n-1}_{(n)})$ take negative values, hence it suffices to check that for all $\varepsilon\gt 0$, $$ \mathbb P\left(Y_n\leqslant -\varepsilon\right)\to 0. $$ Using the distribution function you got, we are reduced to show that $$ \lim_{n\to +\infty}\left(1-\exp(-n\varepsilon)\right)^{\frac{n}{n-1}} =1 $$ or equivalently, taking the $\log$, that $\ln\left(1-\exp(-n\varepsilon)\right)\to 0$.