Definition:Limit Inferior
Definition
Let $\sequence {x_n}$ be a bounded sequence in $\R$.
Definition 1
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:
- $\ds \map {\liminf_{n \mathop \to \infty} } {x_n} = \underline l$
Definition 2
The limit inferior of $\sequence {x_n}$ is defined and denoted as:
- $\ds \map {\liminf_{n \mathop \to \infty} } {x_n} = \sup \set {\inf_{m \mathop \ge n} x_m: n \in \N}$
Also known as
The limit inferior is also called the lower limit, or just liminf.
However, note that the term lower limit has a subtly different meaning in the context of topology, so to avoid ambiguity its use here is not recommended.
Examples
Sequence of Reciprocals
Let $\sequence {a_n}$ be the sequence defined as:
- $\forall n \in \N_{>0}: a_n = \dfrac 1 n$
The limit inferior of $\sequence {a_n}$ is given by:
- $\ds \map {\liminf_{n \mathop \to \infty} } {a_n} = 0$
Divergent Sequence $\paren {-1}^n$
Let $\sequence {a_n}$ be the sequence defined as:
- $\forall n \in \N_{>0}: a_n = \paren {-1}^n$
The limit inferior of $\sequence {a_n}$ is given by:
- $\ds \map {\liminf_{n \mathop \to \infty} } {a_n} = -1$
Farey Sequence
Consider the Farey sequence:
- $\sequence {a_n} = \dfrac 1 2, \dfrac 1 3, \dfrac 2 3, \dfrac 1 4, \dfrac 2 4, \dfrac 3 4, \dfrac 1 5, \dfrac 2 5, \dfrac 3 5, \dfrac 4 5, \dfrac 1 6, \ldots$
The limit inferior of $\sequence {a_n}$ is given by:
- $\ds \map {\liminf_{n \mathop \to \infty} } {a_n} = 0$
Also see
- Definition:Limit Inferior of Sequence of Sets for an extension of this concept into the field of measure theory.
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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
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): limit inferior