Definition:Limit Inferior/Definition 1

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Let $\sequence {x_n}$ be a bounded sequence in $\R$.

Let $L$ be the set of all real numbers which are the limit of some subsequence of $\sequence {x_n}$.

From Existence of Maximum and Minimum of Bounded Sequence, $L$ has a minimum.

This minimum is called the limit inferior.

It can be denoted:

$\ds \map {\liminf_{n \mathop \to \infty} } {x_n} = \underline l$

Also see

Linguistic Note

The plural of limit inferior is limits inferior.

This is because limit is the noun and inferior is the adjective qualifying that noun.