I intend to post this for a Borel-Cantelli lemma exercise on Math.SE.

The target event is ${\exists i_0 \in \Bbb{N} : \forall i \ge i_0, X_i = 1}$, whose complement is $${\forall i_0 \in \Bbb{N} : \exists i \ge i_0, X_i = 0} = \limsup_i {X_i = 0}.$$

To apply Borel-Cantelli, one has to determine whether $\sum_i P(X_i = 0)<+\infty$.