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I've tried to solve the following problem, but so far I have nothing. Here it goes:

(A Probability Path, Resnick, 5th edition) Suppose $\{B_n, n \geq 1\}$ is a sequence of events such that for som $\delta > 0$, $P(B_n) \geq \delta$, for all $n\geq 1$. Show $\lim \sup_{n\to \infty} B_n \neq \emptyset$. Use this to help show with minimum calculation that in an inifite sequence of independent Bernoulli trials, there is an infinte of successes with probability one.

Besides elementary math theory, so far the book has shown the topics on set theory, probability spces, random variables and independence.

I searched for an answer here, but I found nothing. If this question is repeated, please report it and post the link with the answer. Thank you very much in advance.

Ignacio Correa
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1 Answers1

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This fact is immediate consequence of "continuity from above" property of a measure: $$\limsup_{n\to\infty} B_n=\bigcap_{n=1}^{\infty}\bigcup_{k\geq n} B_k=\bigcap_{n=1}^{\infty} A_n,$$ where $A_1\supseteq A_2\supseteq A_3\ldots$.

If we assume that $\limsup_{n\to\infty} B_n=\bigcap_{n=1}^{\infty} A_n=\emptyset$, then $P(A_n)\to P(\emptyset)=0$. But $P(A_n)\geq P(B_n)\geq \delta$ for all $n$. Contradiction.

For the second question, use Borel-Cantelli lemma for independent events.

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