Sequences, series, summability Definition

Limit superior

Definition

Let $(a_n)$ be a bounded sequence of real numbers. For each $n \in \mathbb{N}$ , let $s_n = \sup \{a_k : k \ge n\}$ . The sequence $(s_n)$ is decreasing and bounded below. The limit superior of $(a_n)$ is $\limsup_{n \to \infty} a_n = \lim_{n \to \infty} s_n = \inf_{n \in \mathbb{N}} \sup_{k \ge n} a_k.$

Remarks

The limit exists because a decreasing sequence bounded below converges. For a sequence that is not bounded above, one sets $\limsup a_n = +\infty$ . For a sequence that is bounded above and not below, the same formula is used in the extended real line, and it can give $-\infty$ . For $a_n = (-1)^n (1 + 1/(n+1))$ , the limit superior is $1$ , although no term equals $1$ . For a bounded sequence, it is the largest cluster point.

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Sequence definition
Real sequence definition
Bounded sequence definition
Limit superior definition

Tag	`0P8BA7T7S5`
URI	`m2://0P8BA7T7S5`
Aliases	`limsup`
Kind	definition
Writing	W0 stub
MSC	40A05
Defines	Limit superior
Uses	Bounded sequence

Limit superior

Definition

Remarks

Reading path

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