# Exact Real Bounds, Leastness, Greatestness and Unattained Endpoints This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed. This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately. batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations · 2026-10-14 · 100 questions This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt. Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged. [Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json) ## 1. Finite Nonempty Real Sets Attain Both Their Bounds The ordered rational values=[1/2, -2/3, 3/4] 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). View answer and explanation([1/2, -2/3, 3/4], 3/4, -2/3, Yes, Yes, finite_nonempty_set_extrema) Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([1/2, -2/3, 3/4], 3/4, -2/3, Yes, Yes, finite_nonempty_set_extrema). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0001/) Supremum and infimum · qf-mod38-supinf-finite-extrema · candidate ## 2. Finite Nonempty Real Sets Attain Both Their Bounds The ordered rational values=[2/3, -3/4, 5/5] 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). View answer and explanation([2/3, -3/4, 1], 1, -3/4, Yes, Yes, finite_nonempty_set_extrema) Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([2/3, -3/4, 1], 1, -3/4, Yes, Yes, finite_nonempty_set_extrema). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0002/) Supremum and infimum · qf-mod38-supinf-finite-extrema · candidate ## 3. Finite Nonempty Real Sets Attain Both Their Bounds 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). View answer and explanation([3/4, -4/5, 7/6], 7/6, -4/5, Yes, Yes, finite_nonempty_set_extrema) Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([3/4, -4/5, 7/6], 7/6, -4/5, Yes, Yes, finite_nonempty_set_extrema). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0003/) Supremum and infimum · qf-mod38-supinf-finite-extrema · candidate ## 4. Finite Nonempty Real Sets Attain Both Their Bounds The ordered rational values=[4/5, -5/6, 9/7] 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). View answer and explanation([4/5, -5/6, 9/7], 9/7, -5/6, Yes, Yes, finite_nonempty_set_extrema) Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([4/5, -5/6, 9/7], 9/7, -5/6, Yes, Yes, finite_nonempty_set_extrema). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0004/) Supremum and infimum · qf-mod38-supinf-finite-extrema · candidate ## 5. Finite Nonempty Real Sets Attain Both Their Bounds The ordered rational values=[5/6, -6/7, 11/8] 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). View answer and explanation([5/6, -6/7, 11/8], 11/8, -6/7, Yes, Yes, finite_nonempty_set_extrema) Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([5/6, -6/7, 11/8], 11/8, -6/7, Yes, Yes, finite_nonempty_set_extrema). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0005/) Supremum and infimum · qf-mod38-supinf-finite-extrema · candidate ## 6. Finite Nonempty Real Sets Attain Both Their Bounds The ordered rational values=[6/7, -7/8, 13/9] 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). View answer and explanation([6/7, -7/8, 13/9], 13/9, -7/8, Yes, Yes, finite_nonempty_set_extrema) Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([6/7, -7/8, 13/9], 13/9, -7/8, Yes, Yes, finite_nonempty_set_extrema). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0006/) Supremum and infimum · qf-mod38-supinf-finite-extrema · candidate ## 7. Finite Nonempty Real Sets Attain Both Their Bounds The ordered rational values=[7/8, -8/9, 15/10] 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). View answer and explanation([7/8, -8/9, 3/2], 3/2, -8/9, Yes, Yes, finite_nonempty_set_extrema) Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([7/8, -8/9, 3/2], 3/2, -8/9, Yes, Yes, finite_nonempty_set_extrema). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0007/) Supremum and infimum · qf-mod38-supinf-finite-extrema · candidate ## 8. Finite Nonempty Real Sets Attain Both Their Bounds The ordered rational values=[8/9, -9/10, 17/11] 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). View answer and explanation([8/9, -9/10, 17/11], 17/11, -9/10, Yes, Yes, finite_nonempty_set_extrema) Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([8/9, -9/10, 17/11], 17/11, -9/10, Yes, Yes, finite_nonempty_set_extrema). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0008/) Supremum and infimum · qf-mod38-supinf-finite-extrema · candidate ## 9. Finite Nonempty Real Sets Attain Both Their Bounds The ordered rational values=[9/10, -10/11, 19/12] 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). View answer and explanation([9/10, -10/11, 19/12], 19/12, -10/11, Yes, Yes, finite_nonempty_set_extrema) Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([9/10, -10/11, 19/12], 19/12, -10/11, Yes, Yes, finite_nonempty_set_extrema). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0009/) Supremum and infimum · qf-mod38-supinf-finite-extrema · candidate ## 10. Finite Nonempty Real Sets Attain Both Their Bounds The ordered rational values=[10/11, -11/12, 21/13] 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). View answer and explanation([10/11, -11/12, 21/13], 21/13, -11/12, Yes, Yes, finite_nonempty_set_extrema) Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([10/11, -11/12, 21/13], 21/13, -11/12, Yes, Yes, finite_nonempty_set_extrema). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0010/) Supremum and infimum · qf-mod38-supinf-finite-extrema · candidate ## 11. Finite Nonempty Real Sets Attain Both Their Bounds The ordered rational values=[11/12, -12/13, 23/14] 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). View answer and explanation([11/12, -12/13, 23/14], 23/14, -12/13, Yes, Yes, finite_nonempty_set_extrema) Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([11/12, -12/13, 23/14], 23/14, -12/13, Yes, Yes, finite_nonempty_set_extrema). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0011/) Supremum and infimum · qf-mod38-supinf-finite-extrema · candidate ## 12. Finite Nonempty Real Sets Attain Both Their Bounds The ordered rational values=[12/13, -13/14, 25/15] 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). View answer and explanation([12/13, -13/14, 5/3], 5/3, -13/14, Yes, Yes, finite_nonempty_set_extrema) Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([12/13, -13/14, 5/3], 5/3, -13/14, Yes, Yes, finite_nonempty_set_extrema). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0012/) Supremum and infimum · qf-mod38-supinf-finite-extrema · candidate ## 13. Finite Nonempty Real Sets Attain Both Their Bounds The ordered rational values=[13/14, -14/15, 27/16] 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). View answer and explanation([13/14, -14/15, 27/16], 27/16, -14/15, Yes, Yes, finite_nonempty_set_extrema) Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([13/14, -14/15, 27/16], 27/16, -14/15, Yes, Yes, finite_nonempty_set_extrema). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0013/) Supremum and infimum · qf-mod38-supinf-finite-extrema · candidate ## 14. Finite Nonempty Real Sets Attain Both Their Bounds The ordered rational values=[14/15, -15/16, 29/17] 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). View answer and explanation([14/15, -15/16, 29/17], 29/17, -15/16, Yes, Yes, finite_nonempty_set_extrema) Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([14/15, -15/16, 29/17], 29/17, -15/16, Yes, Yes, finite_nonempty_set_extrema). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0014/) Supremum and infimum · qf-mod38-supinf-finite-extrema · candidate ## 15. Finite Nonempty Real Sets Attain Both Their Bounds The ordered rational values=[15/16, -16/17, 31/18] 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). View answer and explanation([15/16, -16/17, 31/18], 31/18, -16/17, Yes, Yes, finite_nonempty_set_extrema) Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([15/16, -16/17, 31/18], 31/18, -16/17, Yes, Yes, finite_nonempty_set_extrema). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0015/) Supremum and infimum · qf-mod38-supinf-finite-extrema · candidate ## 16. Finite Nonempty Real Sets Attain Both Their Bounds The ordered rational values=[16/17, -17/18, 33/19] 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). View answer and explanation([16/17, -17/18, 33/19], 33/19, -17/18, Yes, Yes, finite_nonempty_set_extrema) Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([16/17, -17/18, 33/19], 33/19, -17/18, Yes, Yes, finite_nonempty_set_extrema). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0016/) Supremum and infimum · qf-mod38-supinf-finite-extrema · candidate ## 17. Finite Nonempty Real Sets Attain Both Their Bounds The ordered rational values=[17/18, -18/19, 35/20] 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). View answer and explanation([17/18, -18/19, 7/4], 7/4, -18/19, Yes, Yes, finite_nonempty_set_extrema) Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([17/18, -18/19, 7/4], 7/4, -18/19, Yes, Yes, finite_nonempty_set_extrema). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0017/) Supremum and infimum · qf-mod38-supinf-finite-extrema · candidate ## 18. Finite Nonempty Real Sets Attain Both Their Bounds The ordered rational values=[18/19, -19/20, 37/21] 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). View answer and explanation([18/19, -19/20, 37/21], 37/21, -19/20, Yes, Yes, finite_nonempty_set_extrema) Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([18/19, -19/20, 37/21], 37/21, -19/20, Yes, Yes, finite_nonempty_set_extrema). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0018/) Supremum and infimum · qf-mod38-supinf-finite-extrema · candidate ## 19. Finite Nonempty Real Sets Attain Both Their Bounds The ordered rational values=[19/20, -20/21, 39/22] 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). View answer and explanation([19/20, -20/21, 39/22], 39/22, -20/21, Yes, Yes, finite_nonempty_set_extrema) Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([19/20, -20/21, 39/22], 39/22, -20/21, Yes, Yes, finite_nonempty_set_extrema). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0019/) Supremum and infimum · qf-mod38-supinf-finite-extrema · candidate ## 20. Finite Nonempty Real Sets Attain Both Their Bounds The ordered rational values=[20/21, -21/22, 41/23] 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). View answer and explanation([20/21, -21/22, 41/23], 41/23, -21/22, Yes, Yes, finite_nonempty_set_extrema) Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([20/21, -21/22, 41/23], 41/23, -21/22, Yes, Yes, finite_nonempty_set_extrema). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0020/) Supremum and infimum · qf-mod38-supinf-finite-extrema · candidate ## 21. Finite Nonempty Real Sets Attain Both Their Bounds The ordered rational values=[21/22, -22/23, 43/24] 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). View answer and explanation([21/22, -22/23, 43/24], 43/24, -22/23, Yes, Yes, finite_nonempty_set_extrema) Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([21/22, -22/23, 43/24], 43/24, -22/23, Yes, Yes, finite_nonempty_set_extrema). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0021/) Supremum and infimum · qf-mod38-supinf-finite-extrema · candidate ## 22. Finite Nonempty Real Sets Attain Both Their Bounds The ordered rational values=[22/23, -23/24, 45/25] 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). View answer and explanation([22/23, -23/24, 9/5], 9/5, -23/24, Yes, Yes, finite_nonempty_set_extrema) Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([22/23, -23/24, 9/5], 9/5, -23/24, Yes, Yes, finite_nonempty_set_extrema). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0022/) Supremum and infimum · qf-mod38-supinf-finite-extrema · candidate ## 23. Finite Nonempty Real Sets Attain Both Their Bounds The ordered rational values=[23/24, -24/25, 47/26] 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). View answer and explanation([23/24, -24/25, 47/26], 47/26, -24/25, Yes, Yes, finite_nonempty_set_extrema) Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([23/24, -24/25, 47/26], 47/26, -24/25, Yes, Yes, finite_nonempty_set_extrema). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0023/) Supremum and infimum · qf-mod38-supinf-finite-extrema · candidate ## 24. Finite Nonempty Real Sets Attain Both Their Bounds The ordered rational values=[24/25, -25/26, 49/27] 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). View answer and explanation([24/25, -25/26, 49/27], 49/27, -25/26, Yes, Yes, finite_nonempty_set_extrema) Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([24/25, -25/26, 49/27], 49/27, -25/26, Yes, Yes, finite_nonempty_set_extrema). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0024/) Supremum and infimum · qf-mod38-supinf-finite-extrema · candidate ## 25. Finite Nonempty Real Sets Attain Both Their Bounds The ordered rational values=[25/26, -26/27, 51/28] 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). View answer and explanation([25/26, -26/27, 51/28], 51/28, -26/27, Yes, Yes, finite_nonempty_set_extrema) Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([25/26, -26/27, 51/28], 51/28, -26/27, Yes, Yes, finite_nonempty_set_extrema). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0025/) Supremum and infimum · qf-mod38-supinf-finite-extrema · candidate ## 26. An Upper Bound Still Needs Leastness to Be the Supremum The ordered rational values=[1/2, -2/3, 3/4] 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. Candidate=3/4. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary). View answer and explanation(3/4, Yes, 3/4, Yes, upper_bound_and_leastness_are_separate) Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (3/4, Yes, 3/4, Yes, upper_bound_and_leastness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0026/) Supremum and infimum · qf-mod38-supinf-upper-bound-leastness · candidate ## 27. An Upper Bound Still Needs Leastness to Be the Supremum The ordered rational values=[2/3, -3/4, 5/5] 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. Candidate=1/1. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary). View answer and explanation(1, Yes, 1, Yes, upper_bound_and_leastness_are_separate) Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (1, Yes, 1, Yes, upper_bound_and_leastness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0027/) Supremum and infimum · qf-mod38-supinf-upper-bound-leastness · candidate ## 28. An Upper Bound Still Needs Leastness to Be the Supremum 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. Candidate=7/6. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary). View answer and explanation(7/6, Yes, 7/6, Yes, upper_bound_and_leastness_are_separate) Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (7/6, Yes, 7/6, Yes, upper_bound_and_leastness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0028/) Supremum and infimum · qf-mod38-supinf-upper-bound-leastness · candidate ## 29. An Upper Bound Still Needs Leastness to Be the Supremum The ordered rational values=[4/5, -5/6, 9/7] 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. Candidate=9/7. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary). View answer and explanation(9/7, Yes, 9/7, Yes, upper_bound_and_leastness_are_separate) Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (9/7, Yes, 9/7, Yes, upper_bound_and_leastness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0029/) Supremum and infimum · qf-mod38-supinf-upper-bound-leastness · candidate ## 30. An Upper Bound Still Needs Leastness to Be the Supremum The ordered rational values=[5/6, -6/7, 11/8] 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. Candidate=11/8. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary). View answer and explanation(11/8, Yes, 11/8, Yes, upper_bound_and_leastness_are_separate) Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (11/8, Yes, 11/8, Yes, upper_bound_and_leastness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0030/) Supremum and infimum · qf-mod38-supinf-upper-bound-leastness · candidate ## 31. An Upper Bound Still Needs Leastness to Be the Supremum The ordered rational values=[6/7, -7/8, 13/9] 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. Candidate=13/9. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary). View answer and explanation(13/9, Yes, 13/9, Yes, upper_bound_and_leastness_are_separate) Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (13/9, Yes, 13/9, Yes, upper_bound_and_leastness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0031/) Supremum and infimum · qf-mod38-supinf-upper-bound-leastness · candidate ## 32. An Upper Bound Still Needs Leastness to Be the Supremum The ordered rational values=[7/8, -8/9, 15/10] 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. Candidate=3/2. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary). View answer and explanation(3/2, Yes, 3/2, Yes, upper_bound_and_leastness_are_separate) Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (3/2, Yes, 3/2, Yes, upper_bound_and_leastness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0032/) Supremum and infimum · qf-mod38-supinf-upper-bound-leastness · candidate ## 33. An Upper Bound Still Needs Leastness to Be the Supremum The ordered rational values=[8/9, -9/10, 17/11] 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. Candidate=17/11. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary). View answer and explanation(17/11, Yes, 17/11, Yes, upper_bound_and_leastness_are_separate) Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (17/11, Yes, 17/11, Yes, upper_bound_and_leastness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0033/) Supremum and infimum · qf-mod38-supinf-upper-bound-leastness · candidate ## 34. An Upper Bound Still Needs Leastness to Be the Supremum The ordered rational values=[9/10, -10/11, 19/12] 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. Candidate=19/12. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary). View answer and explanation(19/12, Yes, 19/12, Yes, upper_bound_and_leastness_are_separate) Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (19/12, Yes, 19/12, Yes, upper_bound_and_leastness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0034/) Supremum and infimum · qf-mod38-supinf-upper-bound-leastness · candidate ## 35. An Upper Bound Still Needs Leastness to Be the Supremum The ordered rational values=[10/11, -11/12, 21/13] 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. Candidate=21/13. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary). View answer and explanation(21/13, Yes, 21/13, Yes, upper_bound_and_leastness_are_separate) Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (21/13, Yes, 21/13, Yes, upper_bound_and_leastness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0035/) Supremum and infimum · qf-mod38-supinf-upper-bound-leastness · candidate ## 36. An Upper Bound Still Needs Leastness to Be the Supremum The ordered rational values=[11/12, -12/13, 23/14] 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. Candidate=23/14. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary). View answer and explanation(23/14, Yes, 23/14, Yes, upper_bound_and_leastness_are_separate) Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (23/14, Yes, 23/14, Yes, upper_bound_and_leastness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0036/) Supremum and infimum · qf-mod38-supinf-upper-bound-leastness · candidate ## 37. An Upper Bound Still Needs Leastness to Be the Supremum The ordered rational values=[12/13, -13/14, 25/15] 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. Candidate=5/3. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary). View answer and explanation(5/3, Yes, 5/3, Yes, upper_bound_and_leastness_are_separate) Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (5/3, Yes, 5/3, Yes, upper_bound_and_leastness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0037/) Supremum and infimum · qf-mod38-supinf-upper-bound-leastness · candidate ## 38. An Upper Bound Still Needs Leastness to Be the Supremum The ordered rational values=[13/14, -14/15, 27/16] 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. Candidate=27/16. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary). View answer and explanation(27/16, Yes, 27/16, Yes, upper_bound_and_leastness_are_separate) Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (27/16, Yes, 27/16, Yes, upper_bound_and_leastness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0038/) Supremum and infimum · qf-mod38-supinf-upper-bound-leastness · candidate ## 39. An Upper Bound Still Needs Leastness to Be the Supremum The ordered rational values=[14/15, -15/16, 29/17] 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. Candidate=29/17. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary). View answer and explanation(29/17, Yes, 29/17, Yes, upper_bound_and_leastness_are_separate) Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (29/17, Yes, 29/17, Yes, upper_bound_and_leastness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0039/) Supremum and infimum · qf-mod38-supinf-upper-bound-leastness · candidate ## 40. An Upper Bound Still Needs Leastness to Be the Supremum The ordered rational values=[15/16, -16/17, 31/18] 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. Candidate=31/18. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary). View answer and explanation(31/18, Yes, 31/18, Yes, upper_bound_and_leastness_are_separate) Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (31/18, Yes, 31/18, Yes, upper_bound_and_leastness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0040/) Supremum and infimum · qf-mod38-supinf-upper-bound-leastness · candidate ## 41. An Upper Bound Still Needs Leastness to Be the Supremum The ordered rational values=[16/17, -17/18, 33/19] 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. Candidate=33/19. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary). View answer and explanation(33/19, Yes, 33/19, Yes, upper_bound_and_leastness_are_separate) Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (33/19, Yes, 33/19, Yes, upper_bound_and_leastness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0041/) Supremum and infimum · qf-mod38-supinf-upper-bound-leastness · candidate ## 42. An Upper Bound Still Needs Leastness to Be the Supremum The ordered rational values=[17/18, -18/19, 35/20] 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. Candidate=7/4. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary). View answer and explanation(7/4, Yes, 7/4, Yes, upper_bound_and_leastness_are_separate) Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (7/4, Yes, 7/4, Yes, upper_bound_and_leastness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0042/) Supremum and infimum · qf-mod38-supinf-upper-bound-leastness · candidate ## 43. An Upper Bound Still Needs Leastness to Be the Supremum The ordered rational values=[18/19, -19/20, 37/21] 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. Candidate=37/21. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary). View answer and explanation(37/21, Yes, 37/21, Yes, upper_bound_and_leastness_are_separate) Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (37/21, Yes, 37/21, Yes, upper_bound_and_leastness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0043/) Supremum and infimum · qf-mod38-supinf-upper-bound-leastness · candidate ## 44. An Upper Bound Still Needs Leastness to Be the Supremum The ordered rational values=[19/20, -20/21, 39/22] 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. Candidate=39/22. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary). View answer and explanation(39/22, Yes, 39/22, Yes, upper_bound_and_leastness_are_separate) Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (39/22, Yes, 39/22, Yes, upper_bound_and_leastness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0044/) Supremum and infimum · qf-mod38-supinf-upper-bound-leastness · candidate ## 45. An Upper Bound Still Needs Leastness to Be the Supremum The ordered rational values=[20/21, -21/22, 41/23] 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. Candidate=41/23. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary). View answer and explanation(41/23, Yes, 41/23, Yes, upper_bound_and_leastness_are_separate) Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (41/23, Yes, 41/23, Yes, upper_bound_and_leastness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0045/) Supremum and infimum · qf-mod38-supinf-upper-bound-leastness · candidate ## 46. An Upper Bound Still Needs Leastness to Be the Supremum The ordered rational values=[21/22, -22/23, 43/24] 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. Candidate=43/24. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary). View answer and explanation(43/24, Yes, 43/24, Yes, upper_bound_and_leastness_are_separate) Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (43/24, Yes, 43/24, Yes, upper_bound_and_leastness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0046/) Supremum and infimum · qf-mod38-supinf-upper-bound-leastness · candidate ## 47. An Upper Bound Still Needs Leastness to Be the Supremum The ordered rational values=[22/23, -23/24, 45/25] 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. Candidate=9/5. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary). View answer and explanation(9/5, Yes, 9/5, Yes, upper_bound_and_leastness_are_separate) Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (9/5, Yes, 9/5, Yes, upper_bound_and_leastness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0047/) Supremum and infimum · qf-mod38-supinf-upper-bound-leastness · candidate ## 48. An Upper Bound Still Needs Leastness to Be the Supremum The ordered rational values=[23/24, -24/25, 47/26] 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. Candidate=47/26. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary). View answer and explanation(47/26, Yes, 47/26, Yes, upper_bound_and_leastness_are_separate) Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (47/26, Yes, 47/26, Yes, upper_bound_and_leastness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0048/) Supremum and infimum · qf-mod38-supinf-upper-bound-leastness · candidate ## 49. An Upper Bound Still Needs Leastness to Be the Supremum The ordered rational values=[24/25, -25/26, 49/27] 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. Candidate=49/27. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary). View answer and explanation(49/27, Yes, 49/27, Yes, upper_bound_and_leastness_are_separate) Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (49/27, Yes, 49/27, Yes, upper_bound_and_leastness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0049/) Supremum and infimum · qf-mod38-supinf-upper-bound-leastness · candidate ## 50. An Upper Bound Still Needs Leastness to Be the Supremum The ordered rational values=[25/26, -26/27, 51/28] 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. Candidate=51/28. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary). View answer and explanation(51/28, Yes, 51/28, Yes, upper_bound_and_leastness_are_separate) Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (51/28, Yes, 51/28, Yes, upper_bound_and_leastness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0050/) Supremum and infimum · qf-mod38-supinf-upper-bound-leastness · candidate ## 51. A Lower Bound Still Needs Greatestness to Be the Infimum The ordered rational values=[1/2, -2/3, 3/4] 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. Candidate=-2/3. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary). View answer and explanation(-2/3, Yes, -2/3, Yes, lower_bound_and_greatestness_are_separate) Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-2/3, Yes, -2/3, Yes, lower_bound_and_greatestness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0051/) Supremum and infimum · qf-mod38-supinf-lower-bound-greatestness · candidate ## 52. A Lower Bound Still Needs Greatestness to Be the Infimum The ordered rational values=[2/3, -3/4, 5/5] 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. Candidate=-3/4. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary). View answer and explanation(-3/4, Yes, -3/4, Yes, lower_bound_and_greatestness_are_separate) Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-3/4, Yes, -3/4, Yes, lower_bound_and_greatestness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0052/) Supremum and infimum · qf-mod38-supinf-lower-bound-greatestness · candidate ## 53. A Lower Bound Still Needs Greatestness to Be the Infimum 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. Candidate=-4/5. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary). View answer and explanation(-4/5, Yes, -4/5, Yes, lower_bound_and_greatestness_are_separate) Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-4/5, Yes, -4/5, Yes, lower_bound_and_greatestness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0053/) Supremum and infimum · qf-mod38-supinf-lower-bound-greatestness · candidate ## 54. A Lower Bound Still Needs Greatestness to Be the Infimum The ordered rational values=[4/5, -5/6, 9/7] 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. Candidate=-5/6. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary). View answer and explanation(-5/6, Yes, -5/6, Yes, lower_bound_and_greatestness_are_separate) Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-5/6, Yes, -5/6, Yes, lower_bound_and_greatestness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0054/) Supremum and infimum · qf-mod38-supinf-lower-bound-greatestness · candidate ## 55. A Lower Bound Still Needs Greatestness to Be the Infimum The ordered rational values=[5/6, -6/7, 11/8] 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. Candidate=-6/7. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary). View answer and explanation(-6/7, Yes, -6/7, Yes, lower_bound_and_greatestness_are_separate) Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-6/7, Yes, -6/7, Yes, lower_bound_and_greatestness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0055/) Supremum and infimum · qf-mod38-supinf-lower-bound-greatestness · candidate ## 56. A Lower Bound Still Needs Greatestness to Be the Infimum The ordered rational values=[6/7, -7/8, 13/9] 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. Candidate=-7/8. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary). View answer and explanation(-7/8, Yes, -7/8, Yes, lower_bound_and_greatestness_are_separate) Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-7/8, Yes, -7/8, Yes, lower_bound_and_greatestness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0056/) Supremum and infimum · qf-mod38-supinf-lower-bound-greatestness · candidate ## 57. A Lower Bound Still Needs Greatestness to Be the Infimum The ordered rational values=[7/8, -8/9, 15/10] 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. Candidate=-8/9. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary). View answer and explanation(-8/9, Yes, -8/9, Yes, lower_bound_and_greatestness_are_separate) Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-8/9, Yes, -8/9, Yes, lower_bound_and_greatestness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0057/) Supremum and infimum · qf-mod38-supinf-lower-bound-greatestness · candidate ## 58. A Lower Bound Still Needs Greatestness to Be the Infimum The ordered rational values=[8/9, -9/10, 17/11] 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. Candidate=-9/10. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary). View answer and explanation(-9/10, Yes, -9/10, Yes, lower_bound_and_greatestness_are_separate) Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-9/10, Yes, -9/10, Yes, lower_bound_and_greatestness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0058/) Supremum and infimum · qf-mod38-supinf-lower-bound-greatestness · candidate ## 59. A Lower Bound Still Needs Greatestness to Be the Infimum The ordered rational values=[9/10, -10/11, 19/12] 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. Candidate=-10/11. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary). View answer and explanation(-10/11, Yes, -10/11, Yes, lower_bound_and_greatestness_are_separate) Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-10/11, Yes, -10/11, Yes, lower_bound_and_greatestness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0059/) Supremum and infimum · qf-mod38-supinf-lower-bound-greatestness · candidate ## 60. A Lower Bound Still Needs Greatestness to Be the Infimum The ordered rational values=[10/11, -11/12, 21/13] 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. Candidate=-11/12. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary). View answer and explanation(-11/12, Yes, -11/12, Yes, lower_bound_and_greatestness_are_separate) Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-11/12, Yes, -11/12, Yes, lower_bound_and_greatestness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0060/) Supremum and infimum · qf-mod38-supinf-lower-bound-greatestness · candidate ## 61. A Lower Bound Still Needs Greatestness to Be the Infimum The ordered rational values=[11/12, -12/13, 23/14] 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. Candidate=-12/13. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary). View answer and explanation(-12/13, Yes, -12/13, Yes, lower_bound_and_greatestness_are_separate) Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-12/13, Yes, -12/13, Yes, lower_bound_and_greatestness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0061/) Supremum and infimum · qf-mod38-supinf-lower-bound-greatestness · candidate ## 62. A Lower Bound Still Needs Greatestness to Be the Infimum The ordered rational values=[12/13, -13/14, 25/15] 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. Candidate=-13/14. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary). View answer and explanation(-13/14, Yes, -13/14, Yes, lower_bound_and_greatestness_are_separate) Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-13/14, Yes, -13/14, Yes, lower_bound_and_greatestness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0062/) Supremum and infimum · qf-mod38-supinf-lower-bound-greatestness · candidate ## 63. A Lower Bound Still Needs Greatestness to Be the Infimum The ordered rational values=[13/14, -14/15, 27/16] 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. Candidate=-14/15. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary). View answer and explanation(-14/15, Yes, -14/15, Yes, lower_bound_and_greatestness_are_separate) Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-14/15, Yes, -14/15, Yes, lower_bound_and_greatestness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0063/) Supremum and infimum · qf-mod38-supinf-lower-bound-greatestness · candidate ## 64. A Lower Bound Still Needs Greatestness to Be the Infimum The ordered rational values=[14/15, -15/16, 29/17] 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. Candidate=-15/16. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary). View answer and explanation(-15/16, Yes, -15/16, Yes, lower_bound_and_greatestness_are_separate) Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-15/16, Yes, -15/16, Yes, lower_bound_and_greatestness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0064/) Supremum and infimum · qf-mod38-supinf-lower-bound-greatestness · candidate ## 65. A Lower Bound Still Needs Greatestness to Be the Infimum The ordered rational values=[15/16, -16/17, 31/18] 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. Candidate=-16/17. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary). View answer and explanation(-16/17, Yes, -16/17, Yes, lower_bound_and_greatestness_are_separate) Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-16/17, Yes, -16/17, Yes, lower_bound_and_greatestness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0065/) Supremum and infimum · qf-mod38-supinf-lower-bound-greatestness · candidate ## 66. A Lower Bound Still Needs Greatestness to Be the Infimum The ordered rational values=[16/17, -17/18, 33/19] 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. Candidate=-17/18. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary). View answer and explanation(-17/18, Yes, -17/18, Yes, lower_bound_and_greatestness_are_separate) Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-17/18, Yes, -17/18, Yes, lower_bound_and_greatestness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0066/) Supremum and infimum · qf-mod38-supinf-lower-bound-greatestness · candidate ## 67. A Lower Bound Still Needs Greatestness to Be the Infimum The ordered rational values=[17/18, -18/19, 35/20] 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. Candidate=-18/19. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary). View answer and explanation(-18/19, Yes, -18/19, Yes, lower_bound_and_greatestness_are_separate) Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-18/19, Yes, -18/19, Yes, lower_bound_and_greatestness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0067/) Supremum and infimum · qf-mod38-supinf-lower-bound-greatestness · candidate ## 68. A Lower Bound Still Needs Greatestness to Be the Infimum The ordered rational values=[18/19, -19/20, 37/21] 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. Candidate=-19/20. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary). View answer and explanation(-19/20, Yes, -19/20, Yes, lower_bound_and_greatestness_are_separate) Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-19/20, Yes, -19/20, Yes, lower_bound_and_greatestness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0068/) Supremum and infimum · qf-mod38-supinf-lower-bound-greatestness · candidate ## 69. A Lower Bound Still Needs Greatestness to Be the Infimum The ordered rational values=[19/20, -20/21, 39/22] 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. Candidate=-20/21. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary). View answer and explanation(-20/21, Yes, -20/21, Yes, lower_bound_and_greatestness_are_separate) Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-20/21, Yes, -20/21, Yes, lower_bound_and_greatestness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0069/) Supremum and infimum · qf-mod38-supinf-lower-bound-greatestness · candidate ## 70. A Lower Bound Still Needs Greatestness to Be the Infimum The ordered rational values=[20/21, -21/22, 41/23] 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. Candidate=-21/22. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary). View answer and explanation(-21/22, Yes, -21/22, Yes, lower_bound_and_greatestness_are_separate) Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-21/22, Yes, -21/22, Yes, lower_bound_and_greatestness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0070/) Supremum and infimum · qf-mod38-supinf-lower-bound-greatestness · candidate ## 71. A Lower Bound Still Needs Greatestness to Be the Infimum The ordered rational values=[21/22, -22/23, 43/24] 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. Candidate=-22/23. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary). View answer and explanation(-22/23, Yes, -22/23, Yes, lower_bound_and_greatestness_are_separate) Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-22/23, Yes, -22/23, Yes, lower_bound_and_greatestness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0071/) Supremum and infimum · qf-mod38-supinf-lower-bound-greatestness · candidate ## 72. A Lower Bound Still Needs Greatestness to Be the Infimum The ordered rational values=[22/23, -23/24, 45/25] 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. Candidate=-23/24. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary). View answer and explanation(-23/24, Yes, -23/24, Yes, lower_bound_and_greatestness_are_separate) Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-23/24, Yes, -23/24, Yes, lower_bound_and_greatestness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0072/) Supremum and infimum · qf-mod38-supinf-lower-bound-greatestness · candidate ## 73. A Lower Bound Still Needs Greatestness to Be the Infimum The ordered rational values=[23/24, -24/25, 47/26] 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. Candidate=-24/25. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary). View answer and explanation(-24/25, Yes, -24/25, Yes, lower_bound_and_greatestness_are_separate) Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-24/25, Yes, -24/25, Yes, lower_bound_and_greatestness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0073/) Supremum and infimum · qf-mod38-supinf-lower-bound-greatestness · candidate ## 74. A Lower Bound Still Needs Greatestness to Be the Infimum The ordered rational values=[24/25, -25/26, 49/27] 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. Candidate=-25/26. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary). View answer and explanation(-25/26, Yes, -25/26, Yes, lower_bound_and_greatestness_are_separate) Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-25/26, Yes, -25/26, Yes, lower_bound_and_greatestness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0074/) Supremum and infimum · qf-mod38-supinf-lower-bound-greatestness · candidate ## 75. A Lower Bound Still Needs Greatestness to Be the Infimum The ordered rational values=[25/26, -26/27, 51/28] 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. Candidate=-26/27. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary). View answer and explanation(-26/27, Yes, -26/27, Yes, lower_bound_and_greatestness_are_separate) Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-26/27, Yes, -26/27, Yes, lower_bound_and_greatestness_are_separate). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r02-supremum-and-infimum-0075/) Supremum and infimum · qf-mod38-supinf-lower-bound-greatestness · candidate ## 76. The Bounds of an Open Interval Are Not Its Maximum or Minimum Use the entire nonempty bounded real open interval (0/1, 2/1), with left