$X_n$ be a sequence of random variables defined on the probability space $(\Omega, \mathcal{F}, P)$. Then $X_n \rightarrow X (a.s.)$ if and only if $\forall \epsilon > 0, P({\omega \in \Omega: |X_n(\omega) - X(\omega)| > \epsilon \ i.o.}) = 0$.
$a.s.$ - almost surely, $i.o.$ - infinitely often
I am trying to prove the above statement in $\Rightarrow$ direction. From the definition of $a.s.$ convergence, $P\{\omega \in \Omega: X_n(\omega) \not \rightarrow X(\omega)\} = 0$
i.e., $P\{\omega \in \Omega: X_n(\omega) \not \rightarrow X(\omega)\} = 0$
i.e., $P\{\omega \in \Omega: lim sup |X_n(\omega) - X(\omega)| > 0\} = 0$
i.e., $P\{\omega \in \Omega: \exists \epsilon(\omega) > 0 \ s.t.\ |X_n(\omega) - X(\omega)| > 0\} = 0$.
How do I bring $i.o.$ events in the above? My understanding about $i.o.$ is based on $lim sup$ of events. I am unable to connect the events above with $limsup$ of sets.
Any proper direction would be very helpful.