Finite maxima and unattained interval suprema

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

-2<1/3<5/2

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

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

u\ge5/2

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

\ell\le-2

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

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

x\in B\iff0<x<1

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

u<1

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

\ell>0

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

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

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

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 obligations

For 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 separately

Call 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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