supremum of a real set

上确界

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

Translation evidence: Literature usage

Meaning in context

For a nonempty real set A bounded above, its supremum is the least real upper bound: it bounds every member of A and is no larger than any other upper bound.

非空且有上界的实数集，其上确界是所有上界中最小的实数；集合无有限上界时，可在扩展实数中记为正无穷。

Example

A=(0,4)A=(0,4) has sup⁡A=4\sup A=4, even though 4∉A4\notin A.

Usage and distinctions

A maximum must belong to the set; a supremum need not be attained.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Michael Taylor — Introduction to Analysis in One Variable§1.6, supremum construction around equation (1.6.35) · 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 · Checked 2026-10-14
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