This is follow up question of Confused by limit superior and limit inferior definition
Definition:
Suppose $\limsup x_n \in \mathbb R$. Then $\beta = \limsup x_n$ if and only if for all $\epsilon > 0$,
(i) $x_n < \beta + \epsilon$ for all except finitely many values of n
(ii) $x_n > \beta - \epsilon$ for infinitely many values of n
My Understanding I completly understand the $x_n < \beta + \epsilon$ and
$x_n > \beta - \epsilon$ thing of the definition.
I take sequences$\left\{ S_{n}\right\} $ {bouded,for sake of simplicity}
on the real line .And two subsequences $\left\{ S_{n_{k}}\right\} $and
$\left\{ S_{n_{k'}}\right\} $,monotonically decreasing and increasing
respectiverly.It becomes obvious $\left\{ S_{n_{k}}\right\} \longrightarrow lower$
bound
and $\left\{ S_{n_{k'}}\right\} $$\longrightarrow$upper bound .Eventually
both sequences will fall in $\epsilon$-neighbourhood of lower bound
and upper bound.
Problem What is meaning of for all except finitely many values of n and for infinitely many values of n
For all except finitely many values of $n$: Means that the set of natural numbers that do not satisfy the property is finite. For example, the set of all numbers $>100$.
For infinitely many values of $n$: Means that the set of natural numbers that satisfy the property is infinite. For example, all even numbers.
Does that answer your question?