Suprema and Infima

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How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

✓ 13 results · all verified · 12 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full by a delegated reviewing agent on the owner's instruction; the judge is an additional, independent cross-model AI review of the proofs. The 1 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

1 · Prerequisites

Real Analysis

Set Theory

2 · Logical flowchart

3 · Items

Definition

Suprema and Infima

3 · Items

Lower bound, bounded below, bounded set

Greatest lower bound (infimum)

Maximum and minimum of a set

Suprema and infima are unique

Reflection through zero exchanges upper and lower bounds

Every nonempty set bounded below has an infimum

Epsilon characterisation of the supremum

Epsilon characterisation of the infimum

The supremum is attained exactly when a maximum exists

Every nonempty finite set of reals has a maximum and a minimum

Monotonicity of the supremum under inclusion

Supremum of a translate: sup(a + S) = a + sup S

Supremum of a scalar multiple

Supremum of a sumset: sup(S + T) = sup S + sup T

Conventions: sup ∅, unbounded sets, and the extended reals

FALSE: the supremum of a set belongs to the set

FALSE: every subset of ℝ has a supremum