Let $a(n)$ be a sequence in $\mathbb R$. The following are equivalent:
$\limsup a(n) = A$;
For every $A' > A$, $a(n) < A'$ for all but finitely many $n$; for every $A'' < A$, $a(n) > A''$ for infinitely many $n$.
So, we have limit superior $A$. As $A'>A$ shouldn't $a(n) < A'$ for infinitely many $n$? As $A'' < A$, $a(n) > A''$ for infinitely many $n$? Does the statement say that if we have a real number that is more than a limit superior then all the sequence for infinitely (in the statement it says finitely) many $n$ are less than that number. And if we have a number that is less than a limit superior then the sequences for infinitely many $n$ are more than that number. What does it mean by equivalence, I can't see any equivalence here.There are only inequalities. Thank you.
just small example
$$a (n)=(-1)^n $$ $$\limsup a_n=1$$
$$b_n=\sin(n\frac {\pi}{3}) $$
$$\limsup b_n=\frac {\sqrt {3}}{2}$$