---
tag: 0P8BA7T7S5
kind: definition
name: Limit superior
aliases: [limsup]
msc: [40A05]
defines: 0T3RZQXDYD
uses: [05FQ3VGAR9]
sources:
  - url: https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior
    locator: The case of sequences of real numbers
---

## Definition

Let $(a_n)$ be a [[bounded sequence|05FQ3VGAR9]] 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|02V2GYAC8P]].
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|0RCKQN1E09]].

## Explanations

### L1 human

The limit superior is the highest value that the sequence keeps coming back to, in the long run.
Each number above it is exceeded by only finitely many terms.