A Follow up question of **Confused by limit superior and limit inferior definition**

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

• You quoted a definition. What is supposed to be defined in that definition? – Hagen von Eitzen Dec 30 '17 at 9:33
• Eventually it can never jump too far above the limsup, staying within epsilon. It can always jump arbitrarily far below the limsup but must always return to the epsilon neighborhood. – jdods Dec 30 '17 at 13:08

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.