# Borell-Cantelli Lemma application

$$A_n=\lbrace X\leq \frac{1}{2}+\frac{1}{n}\rbrace, n=1,2,...$$

(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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