limit superior of a real sequence

数列上极限

Analysis, Measure & PDE · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

The limit superior of a real sequence a_n is the extended-real limit of the decreasing tail suprema sup_{k>=n} a_k.

实数列的上极限是其各尾部上确界构成的单调递减列的扩展实数极限，也可等于正无穷或负无穷。

Example

For an=(−1)n+1/na_n=(-1)^n+1/n, lim sup⁡an=1\limsup a_n=1 and lim inf⁡an=−1\liminf a_n=-1.

Usage and distinctions

It records the eventual upper envelope; it is not the supremum of all terms, which includes early outliers.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Michael Taylor — Introduction to Analysis in One Variable§3.1, equation (3.1.8) · Checked 2026-10-14

Translation sources

  1. 李东风（北京大学）— 金融时间序列分析备课笔记，附录EE.1 集合运算，数列上极限公式 · Checked 2026-10-14
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