Definition:Limit Inferior/Definition 1

Definition

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

• Equivalence of Definitions of Limit Inferior

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.

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 5$: Subsequences: Lim sup and lim inf: $\S 5.13$