Let $X_1,X_2,\cdots$ be i.i.d $p^{\text{th}}$ integrable random variables. Show that $\cfrac{1}{n^{1/p}} \sup_{k\le n} |X_k| \to 0$ almost surely.
Asymptotics of $\max\limits_{1\leqslant k\leqslant n}X_k/n$