The question and a hint are given here. I have a couple of questions about the question and the hint. OP says if $A_m$ is the event that $X_n\leq m$ eventually, then $\cup_m A_m = \{\text{limsup } X_n < \infty\}$. But shouldn't the event $\{\text{limsup } X_n < \infty\}$ mean $X_n\geq m$ eventually for some $m$?
As for the hint, just wanted to confirm that if $B_n$ is the event that $X_n\leq m$ i.o., then $\cup_n B_n = \{\text{liminf } X_n<\infty\}$? Thus $D\supset \{\text{liminf } X_n<\infty \}$ and $1=P(D\cup \{\text{liminf }X_n=\infty\})\leq P(D\cup \{\text{lim }X_n=\infty\}) = 1$? Thanks.
Best Answer
If a non-random sequence $(a_n)$ has a finite $\limsup$, it means that $\lim_{n\to \infty}\sup_{k\geq n}a_k$ is finite (say equal to $M$) hence there exists some $n_0$ such that for all $n\geq n_0$, $\sup_{k\geq n}a_k\leq M+1$ and all the terms are eventually smaller than $m:=M+1$.