# Supremum and infimum

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

For a nonempty appropriately bounded real set, the supremum is its least upper bound and the infimum its greatest lower bound; either need not belong to the set.

mko-supremum-and-infimum · definition · v0.10.0



## Mathematical statement

s=sup A requires s to be an upper bound and no larger than any other upper bound; inf A is defined dually. Complete finite rational sets give an exact slice, and bounded open intervals demonstrate nonattained endpoints. Empty sets, unbounded sets and extended-real infinity are outside this slice.





## Representative formula (exact TeX)

s=sup(A)





## Assumptions


- The ordinary real-valued version uses nonempty sets with the bound required by the chosen extremum.
- All finite values and interval endpoints are exact rationals.





## Symbol scope


- sup — supremum
- inf — infimum





## Intuition, definition, and misconceptions

Being an upper bound and being least are both required.

A supremum or infimum need not be a member.

An attained supremum or infimum is respectively a maximum or minimum.

A supremum is not automatically a largest member of the set.





## Adopted derivation guide

[Finite maxima and unattained interval suprema](https://ocme.evemisslab.com/en/derivations/derivation-mko-supremum-and-infimum-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Real number](https://ocme.evemisslab.com/en/mko/mko-real-number/) — The carrier has real order completeness.
- [Usual order on the real numbers](https://ocme.evemisslab.com/en/mko/mko-order-relation/) — Upper and lower bounds use the order.
- [Set](https://ocme.evemisslab.com/en/mko/mko-set/) — The bounded collection is specified.
- [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/mko/mko-quantifier/) — Leastness and greatestness have universal obligations.





## Verification boundary

Empty sets, unbounded sets and extended reals are not handled here.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add monotone convergence, Bolzano-Weierstrass and the intermediate value theorem.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/supremum-and-infimum.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-supremum-and-infimum/index.md)