Borel Cantelli Exercise 2019

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 $\sum_i P(X_i = 0)<+\infty$.


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