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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

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    $\begingroup$ You quoted a definition. What is supposed to be defined in that definition? $\endgroup$ – Hagen von Eitzen Dec 30 '17 at 9:33
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    $\begingroup$ 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. $\endgroup$ – jdods Dec 30 '17 at 13:08
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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?

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  • $\begingroup$ Very precise translation of such phrases like infinitely many. +1 $\endgroup$ – Paramanand Singh Dec 30 '17 at 12:34

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