For nonnegative RV X s.t. E|X|<$\infty$, $X_n=t_n I_{X>t_n}$ for a sequence of $t_n \to \infty$, since $X_n \to 0$ , then $E(X_n)=t_nP(X>t_n)$ converge to zero as well.
This is from the textbook but I am confused as how we can show $X_n$ converges almost surely to zero. I am wondering this because the textboook used dominated convergence theorem to prove $E(X_n)$ goes to $0$ while we can perfectly easily show $P(X>t_n)$ is going to $0$ as $n \to \infty$ so what is the point of proving $X_n$ converges to $0$ first?
For any $\omega$ in the sample space $X_n (\omega)=0$ for $n$ sufficiently large because $X(\omega) <t_n$ for large $n$. Hence $X_n \to 0$ almost surely . Also $0 \leq X_n \leq X$ by definition so DC T implies $EX_n \to 0$.