Do $\limsup$ and $\liminf$ are the actual bounds of a sequence?

Question

Can you say that $\limsup a_n$ is the upper bound of the sequence $a_n$ and $\liminf a_n$ is the lower bound of $a_n$?

Answer 1

No.

If $a_n=(-1)^n + \frac{(-1)^n}{n}$, then

1. the upper bound of the sequence is $\frac{3}{2}$
2. the lower bound for the sequence is $-2$
3. $\limsup a_n = 1$
4. $\liminf a_n = -1$

so you have a case when the $\limsup$ is not the upper bound, and the $\liminf$ is not the lower bound.