problem 19, section 3.2 book Chung - a course in probability path I can´t prove this problem. Can you help me? The problem says:

If $\{X_n\}$ is a sequence of identically distributed random variables with finite mean, then
$$lim_{n\to\infty}\frac{1}{n}\mathbb{E}\Big[\max_{1\leq j\leq n} |Xj|\Big] = 0$$
[HINT: Use Exercise 17 to express the mean of the maximum.]

In problem 17 show that if X is a positive random variable, then we have
$$\mathbb{E}[X]=\int_0^\infty \!\! P(X &gt; x)\, dx=\int_0^\infty \!\! P(X \geq x)\, dx$$

I prove the problem 19 assuming independence and for the general case I find some upper bounds for the maximum of dependent variables on the internet but none worked.

Thank you for help me.
Use Dominated convergence theorem after applying Exercise 17. The dominating function is P(|X_1|>x).
since h(x)=|f(x)| for any real valued function f, is a convex function(why?) then max(|Xi|) is also convex. by using Jensen's inequality you can prove this. be happy