For random variables, show that $\limsup\limits_nX_n<\infty\Longrightarrow \sup_n X_n<\infty$

Question

Why is the following true ?

$$\limsup\limits_nX_n<\infty\Longrightarrow \sup_n X_n<\infty$$

where, $X_n's$ are random variables.

If we consider only finitely many $X_n$, say $n\in\{1,...,m\}$ then every $X_n$ must have a finite output, since they're random variables, so if we also have $\limsup\limits_nX_n<\infty$, then $X_n$ is finite for all $n$. Is that true ?

Thanks.

EDIT: here is link, where I encountered this problem ($4$th line of the solution).

http://f.666kb.com/i/cr9yzgvnmlryuswod.gif

Answer 1

I don't think this is true in general. $\limsup_n X_n<\infty$ means that $\sup_{k\geq n} X_k$ will converge (and decrease down) to finite number when $n$ large. So this says nothing about $X_n$ for small $n$, some of which may misbehave.

Answer 2

If $Y=\limsup X_n$, then there are only finitely many $i$ such that $X_i<Y+1$.

Specifically, let $Y=\limsup X_n$. Let $N$ be the random variable which is the largest $N$ such that $X_i\geq Y+1$, or $N=0$. Then $\sup X_n <\max(X_1,X_2,\dots,X_N,y+1)$.

(As mentioned above, this assumes that the indexes $i$ of $X_i$ are bounded below and integers. If we are dealing with $\dots,X_{-2},X_{-1},X_0,X_1,\dots$, then this is not true. I assume above that the sequence starts $X_1$.)