limit inferior of a real sequence

数列下极限

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

Translation evidence: Literature usage

Meaning in context

The limit inferior of a real sequence a_n is the extended-real limit of the increasing tail infima inf_{k>=n} a_k.

实数列的下极限是其各尾部下确界构成的单调递增列的扩展实数极限，刻画最终的下方包络。

Example

For $a_n=2+(-1)^n$, the lower limit is $1$ and the upper limit is $3$.

Usage and distinctions

A finite ordinary limit exists exactly when the lower and upper limits coincide at a finite value.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Michael Taylor — Introduction to Analysis in One Variable§3.1, equation (3.1.9) · 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，数列上极限公式后“类似定义下极限” · Checked 2026-10-14

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