Almost sure convergence and i.o. events

Question

$X_n$ 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.

Answer 1

You can show the following inclusion. $\{\omega \in \Omega: X_n(\omega) \not \rightarrow X(\omega)\} \subset \bigcup_{m \geq 1} \{\omega \in \Omega: |X_n(\omega) - X(\omega)| > 1/m \ i.o.\}$ And the use the inequality $$ P(\cup A_m ) \leq \sum_{m}P(A_m).$$

Answer 2

From the definition of convergence, a sequence $a_n$ converges to the limit $L$ iff for all $\epsilon > 0$, $|a_n - L| \le \epsilon$ for all sufficiently large $n$. The latter can't happen if $|a_n - L| > \epsilon$ i.o. Thus $\{\omega: |X_n(\omega) - X(\omega)| > \epsilon \ \text{i.o.}\}$ is disjoint from $\{\omega: X_n(\omega) \to X(\omega)\}$.