Sequences, series, summability Definition

Limit inferior

Definition

Let $(a_n)$ be a bounded sequence of real numbers. For each $n \in \mathbb{N}$ , let $i_n = \inf \{a_k : k \ge n\}$ . The sequence $(i_n)$ is increasing and bounded above. The limit inferior of $(a_n)$ is $\liminf_{n \to \infty} a_n = \lim_{n \to \infty} i_n = \sup_{n \in \mathbb{N}} \inf_{k \ge n} a_k.$

Remarks

The limit exists by the monotone convergence theorem. It satisfies $\liminf a_n = -\limsup (-a_n)$ , where $\limsup$ is the limit superior. Always $\inf_n a_n \le \liminf a_n \le \limsup a_n \le \sup_n a_n$ . For $a_n = (-1)^n$ , the limit inferior is $-1$ and the limit superior is $1$ .

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Sequence definition
Real sequence definition
Bounded sequence definition
Limit inferior definition

Tag	`0H4R3RESGV`
URI	`m2://0H4R3RESGV`
Aliases	`liminf`
Kind	definition
Writing	W0 stub
MSC	40A05
Defines	Limit inferior
Uses	Bounded sequence

Limit inferior

Definition

Remarks

Reading path

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