Borell-Cantelli Lemma application

By definition we have $$ \limsup A_n = \bigcap _{k\ge 1}\bigcup _{n\ge k} A_n = \bigcap _{k\ge 1}\bigcup _{n\ge k}\{ X\le 1/2 +1/n\}=\bigcap _{k\ge 1}\{ X\le 1/2 + 1/k\}=\{X\le 1/2\}$$Note tha the sets $\{X\le a\}$ become larger as $a$ increases, so their union corresponds to the largest $a$ and their intersection corresponds to the smallest $a$. 

We also have $\sum _{n\ge 1} P(A_n)=\sum _{n\ge 1} (1/2 +1/n) = \infty$. So the 1st Borel-Canteli Lemma cannot be used since the series is not finite. As for the 2nd lemma, we can use it if $A_n$s form an independent sequence of events. Suppose $n<m$ then $A_m \subset A_n$ and $$ P(A_n \cap A_m)= P(A_m)=1/2 + 1/m \ne (1/2+1/m)(1/2+1/n)=P(A_n)P(A_m)$$So these events are not independent and the 2nd lemma cannot be used either.