Version of Borel-Cantelli: $P(A_{i} \text{ i.o.)} > 0$

Question

My fellow student stated the following after a lecture about the Borel-Cantelli lemma:

Let $(A_{i})_{i \geq 0}$ be a sequence of disjoint of events on some probabilty space. If $\exists n \in \mathbb{N}, C>1\colon \forall i \geq n, P(A_{i})\geq 1/C$, then $$ P\Bigl(\bigcap^{\infty}_{i=1}\bigcup^{\infty}_{j=i}A_j\Bigr) = P(A_{i} \text{ i.o.}) \geq 1/C^{'} > 0 $$ for some $C'$.

I think that its correct but I can't prove it with the Borel-Cantelli Lemma. Is there some proof with or without Borel-Cantelli?

Answer 1

Such a sequence of events cannot eevn exist. Since the events are disjoint we have $\sum P(A_n) \leq 1$. Convergence of the series $\sum P(A_n)$ implies $P(A_n) \to 0$.

If you drop the hypothesis of disjointness then the result is true. By Fatou's Lemma $P(\lim \inf A_n) \leq \lim \inf P(A_n)$ Taking complements this gives $P(A_{i} i.o.)=P(\lim \sup A_n) \geq \lim \sup P(A_n) \geq 1/C$ so we can taka $C'=C$.