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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!

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    Of course not. Actually, $\Pr(\limsup_{n \rightarrow \infty} E_n) = 1$ means that the events $E_n$ occur for infinitely many $n$s, almost surely. Example to debunk future false claims: if $E_n$ is the event that the $n$th throw is heads in the canonical heads-and-tails setting, then almost surely, $E_n$ happens for infinitely many $n$s and $E_n$ does not happen, also for infinitely many $n$s. – 2017-01-08

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