# Finite maxima and unattained interval suprema

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This is an owner-adopted natural-language teaching guide. Finite/arithmetic checks are not universal proofs; no Lean was run, and review scope is recorded per revision. Original source status labels are retained as historical data. Formulas retain exact TeX; no new semantic AST is invented.

derivation-mko-supremum-and-infimum-01 · ocme-new142-reader-adoption-20261021



## Goal and scope

Find sup/inf of A={-2,1/3,5/2} and B=(0,1), checking bounds and extremality.

Ordinary real values for nonempty bounded sets, excluding empty/unbounded/extended-real cases.





## Givens and symbols


- algebra
- sets
- definition_here


- G1 — A={-2,1/3,5/2} is complete; B={x in R:0<x<1}.


- \sup,\inf — least upper and greatest lower bound
- \max,\min — attained extrema





## Rules and sources


- mko_definition · cited — 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. · mko-supremum-and-infimum
- algebra · prerequisite — Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities.
- sets · prerequisite — Compute unions, intersections, complements, inclusion and preimages element by element.
- definition_here · prerequisite — A supremum is a least upper bound and an infimum a greatest lower bound; it is also a maximum/minimum only when it belongs to the set.





## Main derivation

S01 · impliesBefore

-2<1/3<5/2

After

\forall a\in A:\ -2\le a\le5/2

Conditions: none

Rule: algebra · Uses: G1

Leaf basis: algebra

Ordering all three elements supplies both bounds.



S02 · impliesBefore

u\text{ upper bound of }A,\quad5/2\in A

After

u\ge5/2

Conditions: none

Rule: sets · Uses: G1, S01

Leaf basis: sets

Every upper bound must be at least the attained largest element.



S03 · impliesBefore

\ell\text{ lower bound of }A,\quad-2\in A

After

\ell\le-2

Conditions: none

Rule: sets · Uses: G1, S01

Leaf basis: sets

Every lower bound is no greater than the attained smallest element.



S04 · definitionBefore

5/2\text{ least upper bound},\quad-2\text{ greatest lower bound}

After

\sup A=\max A=5/2,\quad\inf A=\min A=-2

Conditions: none

Rule: definition_here · Uses: S01, S02, S03

Leaf basis: definition_here

These extrema belong to the finite set and are therefore maximum and minimum as well.



S05 · impliesBefore

x\in B\iff0<x<1

After

0\text{ is a lower bound and }1\text{ an upper bound}

Conditions: none

Rule: algebra · Uses: G1

Leaf basis: algebra

The endpoints provide bounds but neither belongs to the open interval.



S06 · impliesBefore

u<1

After

x=(1+\max\{u,0\})/2\in B,\quad x>u

Conditions: u<1

Rule: algebra · Uses: G1, S05

Leaf basis: algebra

The maximum handles every u<1, including negative u, producing a member of B above u; thus u is not an upper bound.



S07 · impliesBefore

\ell>0

After

x=\min\{\ell,1\}/2\in B,\quad x<\ell

Conditions: \ell>0

Rule: algebra · Uses: G1, S05

Leaf basis: algebra

Every positive candidate lower bound is refuted by a smaller member of B.



S08 · definitionBefore

1\text{ least upper bound},\quad0\text{ greatest lower bound},\quad0,1\notin B

After

\sup B=1,\quad\inf B=0;\quad B\text{ has no maximum or minimum}

Conditions: none

Rule: definition_here · Uses: S05, S06, S07

Leaf basis: definition_here

The bounds are not attained; any maximum/minimum would equal the proved unique supremum/infimum, contradicting nonmembership.







## Substeps and lemmas

No substeps in this guide.





## Reverse audits


- S08 · premise_audit · S05, S06, S07 — The fact that the endpoints are not attained is not mistaken for absence of a supremum or infimum.
- S07 · premise_audit · G1, S05 — Both ell≥1 and 0<ell<1 are covered by the minimum formula.
- S06 · premise_audit · G1, S05 — The construction checks 0<x<1 and x>u, rather than a drawing.
- S05 · premise_audit · G1 — Bound inequalities and strict membership inequalities are kept distinct.
- S04 · premise_audit · S01, S02, S03 — Membership was separately checked, not assumed for arbitrary suprema.
- S03 · premise_audit · G1, S01 — The direction is lower bound≤minimum, not the reverse.
- S02 · premise_audit · G1, S01 — Being an upper bound and being least are separate checks.
- S01 · premise_audit · G1 — There are no unlisted members of A.





## Error controls and negative controls

Two obligationsFor each supremum or infimum, verify both that the proposed value is a bound and that it is least or greatest among such bounds.

Check attainment separatelyCall it a maximum/minimum only when membership is established.


- The maximum of B is 1. → reject — 1 is not in B.
- 2 is sup B because it is an upper bound. → reject — It is not least: 1 is a smaller upper bound.
- The empty set has a largest element under the same real convention. → reject — There is no member to attain, and the empty-set version is excluded.





## Result and limitations

A has attained sup=5/2 and inf=-2; B has unattained sup=1 and inf=0.







## Verification and source binding

1 declared checks for this guide; only the listed expressions are covered.


- source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
- adopted source SHA-256: 032fe27da01180f3cc81291f7ffabfb4d24f950b41d70acba163555cd43e319d
- original source SHA-256: 5943c7e129fd8edd4ae73e3d131e0463a489f64dc39ea5e81897df1c99ccccf8
- Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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