Let (left{U_{n}ight}_{n=1}^{infty}) be a sequence of independent and identically distributed UNI(operatorname{FORM}(0,1)) random variables and let (U_{(n)}) be

Question:

Let $\left\{U_{n}ight\}_{n=1}^{\infty}$ be a sequence of independent and identically distributed UNI$\operatorname{FORM}(0,1)$ random variables and let $U_{(n)}$ be the largest order statistic of $U_{1}, \ldots, U_{n}$. That is, $U_{(n)}=\max \left\{U_{1}, \ldots, U_{n}ight\}$. Prove that $U_{(n)} \xrightarrow{\text { a.c. }} 1$ as $n ightarrow \infty$.

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