Supremum and infimum

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

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Real number — The carrier has real order completeness.
• Usual order on the real numbers — Upper and lower bounds use the order.
• Set — The bounded collection is specified.
• Quantifiers and explicit domains — 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.