Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 26G / c

By the union bound,For an $L^2$ random variable, $t^2\mathbb{P}(|X_1|>t)\to 0$. Taking $t=\epsilon \sqrt{n}$ makes the right side tend to zero, proving $M_n/\sqrt{n}\to 0$ in probability.

For any $\epsilon >0$,ThereforeThe last term tends to zero. Taking the limsup and then $\epsilon \downarrow 0$ proves the claim.