By Borel-Cantelli, we know that if $\sum^{\infty}_{n = 1} \Pr(E_n) = \infty$ and the events $(E_n)^{\infty}_{n = 1}$ are independent, then $\Pr(\limsup_{n \rightarrow \infty} E_n) = 1.$
Now, I also read in a manuscript that if $\Pr(\limsup_{n \rightarrow \infty} E_n) = 1$, then the events $E_n$ occur:
"except for finitely many $n$ (almost surely)".
This statement confuses me because I know that $\Pr(\limsup_{n \rightarrow \infty} E_n) = 1$ shouldn't imply that $\Pr(\lim_{n \rightarrow \infty} E_n) = 1$ (almost sure definition) unless it agrees with $\Pr(\liminf_{n \rightarrow \infty} E_n) = 1$. Am I missing something here? Thanks!