TITLE: Definition about "infinitely often"
QUESTION [6 upvotes]: As we know, "$A_n \text{ i.o.}$" means $A_n$ happens infinitely often, where $A_n$ is an event.
I'm not sure whether "the complement of $A_n$ happens for large $n$" is the complement of the preceding event.
Can anyone help me?

REPLY [10 votes]: Assuming you mean "the complement of $A_n$ happens for all sufficiently large $n$," then yes, the following two events are complementary:

$A_n$ happens for infinitely many $n$, and
$\bar{A_n}$ happens for all sufficiently large $n$ (that is, from some point $n_0$ onward.)

Indeed, if $A_n$ happens for infinitely many $n$, then after any given point $n_0$ there will be some $n \ge n_0$ such that $A_n$ happens (and therefore $\bar{A_n}$ does not happen.)
Conversely, if $A_n$ only happens for finitely many $n$, then these $n$'s where $A_n$ happens are bounded by some $n_0$, and for all $n \ge n_0$ the complement $\bar{A_n}$ must happen.
In terms of formal logic, this argument is expressed by the equivalence of the following statements, which correspond to events $\neg$(1) and (2) above, respectively:

$\neg \forall n_0\,\exists n \ge n_0\, A_n$
$\exists n_0\,\forall n \ge n_0\, \neg A_n$.