# Question 20503: Finite Nonempty Real Sets Attain Both Their Bounds [Return to catalogue page 103](https://ocme.evemisslab.com/en/questions/page/103/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations/#q-20261014-advance-20261107-r02-supremum-and-infimum-0003) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0040-20503 · q-20261014-advance-20261107-r02-supremum-and-infimum-0003 ## Original question and givens The ordered rational values=[3/4, -4/5, 7/6] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate). - G001 — The ordered rational values=[3/4, -4/5, 7/6] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-supremum-and-infimum](https://ocme.evemisslab.com/en/mko/mko-supremum-and-infimum/) ## Step-by-step derivation S001 · impliesBefore source operation=supinf-finite-extrema;original snapshot and mathematics retained After explicit ordered source output fields=[values, supremum, infimum, supremum_attained, infimum_attained, certificate] Conditions: none Keep the historical interface marker in the original snapshot while listing source fields here without JavaScript or external lookup.Labels are not mathematical evidence,and No retains its source claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore ORIGINAL complete ordered nonempty finite real-value list=[3/4, -4/5, 7/6] After initialize upper/lower extremum candidates at ORIGINAL first value;duplicates retained and do not change set bounds Conditions: none This is the supplied complete finite set list,not a sample of an unspecified set.Duplicates do not alter bounds but remain in original output order;do not sort the data first. Step identifiers and dependenciesRule: supinf_complete_nonempty_finite_set_not_samples_or_sorting · Uses: G001, S001 S003 · calculationBefore supinf original value1 minus previous maximum := sub(-4/5, 3/4) After supinf original value1 minus previous maximum = -31/20 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore supinf original value1 minus previous minimum := sub(-4/5, 3/4) After supinf original value1 minus previous minimum = -31/20 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore ORIGINAL next index=1;value=-4/5;previous extremum values=[3/4, 3/4];differences=[-31/20, -31/20] After prefix maximum=3/4 attained at original index0;prefix minimum=-4/5 attained at original index1 Conditions: none Compare every original value,preserving prefix maximum/minimum invariants and updating witnesses only for strict improvement.No original value is omitted. Step identifiers and dependenciesRule: supinf_every_value_extremum_prefix_invariant · Uses: G001, S004 S006 · calculationBefore supinf original value2 minus previous maximum := sub(7/6, 3/4) After supinf original value2 minus previous maximum = 5/12 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore supinf original value2 minus previous minimum := sub(7/6, -4/5) After supinf original value2 minus previous minimum = 59/30 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore ORIGINAL next index=2;value=7/6;previous extremum values=[3/4, -4/5];differences=[5/12, 59/30] After prefix maximum=7/6 attained at original index2;prefix minimum=-4/5 attained at original index1 Conditions: none Compare every original value,preserving prefix maximum/minimum invariants and updating witnesses only for strict improvement.No original value is omitted. Step identifiers and dependenciesRule: supinf_every_value_extremum_prefix_invariant · Uses: G001, S007 S009 · impliesBefore ALL original values processed;actual maximum witness=[2, 7/6];actual minimum witness=[1, -4/5] After EVERY listed value between-4/5 and7/6;ANY upper bound must>=actual maximum in the set;ANY lower bound must<=actual minimum;sup=7/6,inf=-4/5,BOTH attained Conditions: none Because the maximum is itself in the set,no upper bound can be smaller;the minimum similarly bounds every lower bound above.This proves both bounding and leastness/greatestness.Attainment uses finiteness and nonemptiness and is not inferred for arbitrary bounded sets. Step identifiers and dependenciesRule: supinf_bounds_and_least_greatest_by_actual_attaining_points · Uses: G001, S008 ## Reverse checks - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Recompute supinf original value2 minus previous minimum with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute supinf original value2 minus previous maximum with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Recompute supinf original value1 minus previous minimum with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute supinf original value1 minus previous maximum with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([3/4, -4/5, 7/6], 7/6, -4/5, Yes, Yes, finite_nonempty_set_extrema) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: c9d7fd812ee191c536557e8c3a24858b8bd12bbd173ff3dd67741113d24301ad - candidate record SHA-256: 6bba2d7cf3482205a0c59777c607796407c9468f40a207f2d25471737987384b - semantic review receipt SHA-256: c6483ed1229af9183fe5c704a2627b1558ad9a31bfbb0d2ee2d5c1a1c299ab4d - question-first witness SHA-256: e81f9a560d630fcb23441146f008c4feca7aceab3ebe4de2cf1997c061375ef7 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0003.json)