infimum of a real set

下确界

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

Translation evidence: Literature usage

Meaning in context

For a nonempty real set A bounded below, its infimum is the greatest real lower bound: it is no larger than any member of A and dominates every other lower bound.

非空且有下界的实数集，其下确界是所有下界中最大的实数；集合无有限下界时，可在扩展实数中记为负无穷。

Example

inf⁡{1+1/n:n≥1}=1\inf\{1+1/n:n\ge1\}=1, which is not a member.

Usage and distinctions

The infimum concerns all lower bounds, whereas a minimum is an attained least member.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Michael Taylor — Introduction to Analysis in One Variable§1.6, infimum definition around equation (1.6.48) · 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，上确界段落末句“类似定义下确界” · Checked 2026-10-14
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