SCHWEITZER 1954 Problem 5

Let $\xi _{1},\xi _{2},\dots ,\xi _{n},...$ be independent random variables of uniform distribution in $(0,1)$ . Show that the distribution of the random variable $\eta _{n}= \sqrt[]{n}\prod_{k=1}^{n}(1-\frac{\xi _{k}}{k}) (n= 1,2,...)$ tends to a limit distribution for $n \to \infty$ . (P. 6)