Clarification on a definition of limsup

Question

My textbook gives the following definition of limit superior, which I'd like some clarification on.

Let $\{s_n\}$ be a sequence that's bounded above and does not diverge to $-\infty$. Then there is a unique real number $\bar{s}$ (the limit superior) such that if $\epsilon > 0$, $$s_n < \bar{s} + \epsilon$$ for "large $n$" and $$s_n > \bar{s} - \epsilon$$ for infinitely many $n$.

First of all, is the textbook saying that for the limit superior to equal $\bar{s}$, the inequalities above must hold for all $\epsilon > 0$? Secondly, what exactly does "large $n$" mean in this context?

Answer 1

Your textbook is not using a very precise language. It should be: $\limsup_ns_n$ is the only real number $\overline s$ such that:

1. for every $\varepsilon>0$, there is a $p\in\mathbb N$ such that$$(\forall n\in\mathbb{N}):n\geqslant p\implies s_n<\overline s+\varepsilon;$$
2. for every $\varepsilon>0$, the inequality $s_n>\overline s-\varepsilon$ holds for infinitely many $n$'s.

Answer 2

"For large $n$", or "eventually", mean "for all $n$ starting at a certain point".

And yes, when the text says "if $\epsilon>0$, then this, then mean that you can do it for every $\epsilon>0$ (the starting point where the inequality holds will change with $\epsilon$, though).

I have always felt that a much better intuition for $\limsup$ is to take it as the supremum of all limits of all subsequences.