LemmaStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)verified 2026-07-26 (claude-opus-5)

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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.

The supremum is attained exactly when a maximum exists

Statement

Let $S \subseteq R$ be nonempty.

If $S$ has a maximum (Maximum and minimum of a set), then $\sup S$ exists and $\sup S = \max S$ .
If $\sup S$ exists and $\sup S \in S$ , then $S$ has a maximum and $\max S = \sup S$ .

Hence, for a set whose supremum exists, the supremum is attained (belongs to the set) precisely when the set has a maximum, and then the two agree.

Facts & Assumptions

Given: A nonempty $S \subseteq R$ .

[L1]

Maximum: $m = \max S$ means $m \in S$ and $s \leq m$ for every $s \in S$ ; a maximum is unique (Maximum and minimum of a set).

[L2]

Supremum: $u = \sup S$ means $u$ is an upper bound of $S$ , that is $s \leq u$ for every $s \in S$ , and $u \leq u^{'}$ for every upper bound $u^{'}$ of $S$ ; it is unique when it exists (Suprema and infima are unique, Complete ordered field (least-upper-bound property)).

Proof

technique · direct

1.1

For claim 1 assume $S$ has a maximum $m$ : then $m \in S$ and $s \leq m$ for every $s \in S$ , so $m$ is in particular an upper bound of $S$ .

assume-hypL1L2

1.2

For claim 2 assume $\sup S$ exists and lies in $S$ , and write $u : = \sup S \in S$ .

assume-hypL2

2.1

Let $u^{'}$ be an arbitrary upper bound of $S$ ; since $m \in S$ , the defining property of an upper bound applied to the element $m$ gives $m \leq u^{'}$ .

step 1.1L2

2.2

Since $u = \sup S$ is an upper bound of $S$ we have $s \leq u$ for every $s \in S$ , and by assumption $u \in S$ ; these are exactly the two requirements for $u$ to be a maximum of $S$ , so $\max S$ exists and equals $u = \sup S$ by uniqueness of the maximum, proving claim 2.

step 1.2L1L2

3.1

Thus $m$ is an upper bound of $S$ with $m \leq u^{'}$ for every upper bound $u^{'}$ of $S$ , which is exactly the definition of a least upper bound; hence $\sup S$ exists and, by uniqueness of the least upper bound, $\sup S = m = \max S$ , proving claim 1.

step 1.1step 2.1L1L2

4.1

Combining the two claims: when $\sup S$ exists, $\sup S \in S$ holds if and only if $S$ has a maximum, and in that case $\sup S = \max S$ .

step 2.2step 3.1∎

Remarks

The dual statement, that $\inf S$ is attained exactly when $S$ has a minimum and then $\inf S = \min S$ , is not proved above. It follows by reflection: $m = \max S$ if and only if $- m = \min (- S)$ , and $\inf X = - \sup (- X)$ (Reflection through zero exchanges upper and lower bounds, Every nonempty set bounded below has an infimum), so applying the two claims proved here to $- S$ and negating gives the minimum form.
Claim 1 needs no completeness assumption: a set with a maximum has a supremum for free, since the maximum is already the least upper bound. Only claim 2 presupposes that $\sup S$ exists, which for a nonempty set bounded above is guaranteed by the least-upper-bound property (Complete ordered field (least-upper-bound property)).
The converse of "the supremum exists" is not "the maximum exists": the set ${x \in R : 0 < x < 1}$ has supremum $1$ and no maximum (FALSE: the supremum of a set belongs to the set). What forces attainment is being nonempty and finite (Every nonempty finite set of reals has a maximum and a minimum); finiteness alone does not, since $\emptyset$ is finite and has no maximum (Maximum and minimum of a set).

Depends on

Used by

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Maximum and minimum (Wikipedia) (standard reference, not scraped)
Infimum and supremum (Wikipedia) (standard reference, not scraped)
David H. Ernst, An Introduction to Proof via Inquiry-Based Learning, Section 5.1 (standard reference, not scraped)