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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- Results about limits inferior can be found here.
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