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.

Answer

Answers can be viewed only if
  1. The questioner was satisfied and accepted the answer, or
  2. The answer was disputed, but the judge evaluated it as 100% correct.
View the answer
The answer is accepted.