Alphabeta Math

Suprema and Infima: Examples and Counterexamples

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✓ 8 results · all verified · 5 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 3 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

Example

sup(0,1) = 1 and inf(0,1) = 0, with neither attained

Example

sup[0,1] = 1 = max[0,1] and inf[0,1] = 0 = min[0,1]

Example

inf{1/n : n ≥ 1} = 0, not attained, while sup{1/n : n ≥ 1} = 1 is

Example

sup{q ∈ ℚ : q > 0, q² < 2} = √2 in ℝ, and no supremum in ℚ

Example

(0,1) + (2,3) = (2,4), with supremum 4 = sup(0,1) + sup(2,3)

Counterexample

A supremum need not belong to its set: sup(0,1) = 1 ∉ (0,1)

Counterexample

An unbounded set has no supremum: the naturals inside ℝ

Counterexample

The empty set is bounded and has no supremum

Sources