Either there exists an extended real number $A$ such that $X_n \to A$ almost surely or the sequence $X_n$ diverge almost surely.

Question

Let $(X_n)$ be a sequence of independent random variables on a probability space.

Then for the sequence $(X_n)$ either there exists an extended real number $A$ such that $X_n \to A$ a.s. or the sequence $X_n$ diverge almost surely.

Need some hints for the problem.

Answer 1

$\{\liminf X_n =\limsup X_n\}$ has probability 0 or 1 by Kolmogorv's 0-1 law. If the probability is 0 then $\{X_n\}$ oscillates with probability 1. If it is 1 then $\liminf X_n =\limsup X_n$ with probability 1 so the sequence converges to a finite limit or to $\infty$ or to $-\infty$ with probability 1.