The book that I am following gives the following definition for limsup:

A real number $\bar{a}$ is said to be the limit superior of a bounded sequence {$a\_n$} iff for each $\epsilon > 0$, the following results hold:

1. $a\_n$ > $\bar{a} - \epsilon$, for infinitely many values of n
2. $\exists$ a positive integer m such that

$a\_n$ < $\bar{a} + \epsilon $ $\forall n \ge m$

I don't understand how 1 and 2 could both be true at the same time. Can someone explain this to me? Also why does the definition mention bounded sequences?