Borell-Cantelli Lemma application
Let $X$ be a uniform random variable on $[0,1]$ and consider the events
$$A_n=\lbrace X\leq \frac{1}{2}+\frac{1}{n}\rbrace, n=1,2,...$$
(a) Compute $\limsup_{n\rightarrow \infty} A_n$.
(I am not great with limsup, so my main question is exacty how to find and justify this answer. My guess is 1/2 but I am not confident.)
(b) Explain why or why not the first Borel-Canteli Lemma can be used to compute $P\lparen \limsup_{n\rightarrow \infty} A_n \rparen=P(A_n i.o.)$ in this example.
(c) Explain why or why not the second Borel-Canteli Lemma can be used to compute $P\lparen \limsup_{n\rightarrow \infty} A_n \rparen=P(A_n i.o.)$ in this example.
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