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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
View answer and explanation
(0, 2, 2, 0, No, No, supremum_and_infimum_not_attained)
The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (0, 2, 2, 0, No, No, supremum_and_infimum_not_attained).
Reviewed step-by-step derivation
Supremum and infimum · qf-mod38-supinf-open-interval-nonattainment · candidate
77. The Bounds of an Open Interval Are Not Its Maximum or Minimum
Use the entire nonempty bounded real open interval (1/2, 3/2), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
View answer and explanation
(1/2, 3/2, 3/2, 1/2, No, No, supremum_and_infimum_not_attained)
The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (1/2, 3/2, 3/2, 1/2, No, No, supremum_and_infimum_not_attained).
Reviewed step-by-step derivation
Supremum and infimum · qf-mod38-supinf-open-interval-nonattainment · candidate
78. The Bounds of an Open Interval Are Not Its Maximum or Minimum
Use the entire nonempty bounded real open interval (2/3, 4/3), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
View answer and explanation
(2/3, 4/3, 4/3, 2/3, No, No, supremum_and_infimum_not_attained)
The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (2/3, 4/3, 4/3, 2/3, No, No, supremum_and_infimum_not_attained).
Reviewed step-by-step derivation
Supremum and infimum · qf-mod38-supinf-open-interval-nonattainment · candidate
79. The Bounds of an Open Interval Are Not Its Maximum or Minimum
Use the entire nonempty bounded real open interval (3/4, 5/4), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
View answer and explanation
(3/4, 5/4, 5/4, 3/4, No, No, supremum_and_infimum_not_attained)
The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (3/4, 5/4, 5/4, 3/4, No, No, supremum_and_infimum_not_attained).
Reviewed step-by-step derivation
Supremum and infimum · qf-mod38-supinf-open-interval-nonattainment · candidate
80. The Bounds of an Open Interval Are Not Its Maximum or Minimum
Use the entire nonempty bounded real open interval (4/5, 6/5), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
View answer and explanation
(4/5, 6/5, 6/5, 4/5, No, No, supremum_and_infimum_not_attained)
The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (4/5, 6/5, 6/5, 4/5, No, No, supremum_and_infimum_not_attained).
Reviewed step-by-step derivation
Supremum and infimum · qf-mod38-supinf-open-interval-nonattainment · candidate
81. The Bounds of an Open Interval Are Not Its Maximum or Minimum
Use the entire nonempty bounded real open interval (5/6, 7/6), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
View answer and explanation
(5/6, 7/6, 7/6, 5/6, No, No, supremum_and_infimum_not_attained)
The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (5/6, 7/6, 7/6, 5/6, No, No, supremum_and_infimum_not_attained).
Reviewed step-by-step derivation
Supremum and infimum · qf-mod38-supinf-open-interval-nonattainment · candidate
82. The Bounds of an Open Interval Are Not Its Maximum or Minimum
Use the entire nonempty bounded real open interval (6/7, 8/7), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
View answer and explanation
(6/7, 8/7, 8/7, 6/7, No, No, supremum_and_infimum_not_attained)
The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (6/7, 8/7, 8/7, 6/7, No, No, supremum_and_infimum_not_attained).
Reviewed step-by-step derivation
Supremum and infimum · qf-mod38-supinf-open-interval-nonattainment · candidate
83. The Bounds of an Open Interval Are Not Its Maximum or Minimum
Use the entire nonempty bounded real open interval (7/8, 9/8), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
View answer and explanation
(7/8, 9/8, 9/8, 7/8, No, No, supremum_and_infimum_not_attained)
The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (7/8, 9/8, 9/8, 7/8, No, No, supremum_and_infimum_not_attained).
Reviewed step-by-step derivation
Supremum and infimum · qf-mod38-supinf-open-interval-nonattainment · candidate
84. The Bounds of an Open Interval Are Not Its Maximum or Minimum
Use the entire nonempty bounded real open interval (8/9, 10/9), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
View answer and explanation
(8/9, 10/9, 10/9, 8/9, No, No, supremum_and_infimum_not_attained)
The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (8/9, 10/9, 10/9, 8/9, No, No, supremum_and_infimum_not_attained).
Reviewed step-by-step derivation
Supremum and infimum · qf-mod38-supinf-open-interval-nonattainment · candidate
85. The Bounds of an Open Interval Are Not Its Maximum or Minimum
Use the entire nonempty bounded real open interval (9/10, 11/10), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
View answer and explanation
(9/10, 11/10, 11/10, 9/10, No, No, supremum_and_infimum_not_attained)
The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (9/10, 11/10, 11/10, 9/10, No, No, supremum_and_infimum_not_attained).
Reviewed step-by-step derivation
Supremum and infimum · qf-mod38-supinf-open-interval-nonattainment · candidate
86. The Bounds of an Open Interval Are Not Its Maximum or Minimum
Use the entire nonempty bounded real open interval (10/11, 12/11), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
View answer and explanation
(10/11, 12/11, 12/11, 10/11, No, No, supremum_and_infimum_not_attained)
The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (10/11, 12/11, 12/11, 10/11, No, No, supremum_and_infimum_not_attained).
Reviewed step-by-step derivation
Supremum and infimum · qf-mod38-supinf-open-interval-nonattainment · candidate
87. The Bounds of an Open Interval Are Not Its Maximum or Minimum
Use the entire nonempty bounded real open interval (11/12, 13/12), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
View answer and explanation
(11/12, 13/12, 13/12, 11/12, No, No, supremum_and_infimum_not_attained)
The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (11/12, 13/12, 13/12, 11/12, No, No, supremum_and_infimum_not_attained).
Reviewed step-by-step derivation
Supremum and infimum · qf-mod38-supinf-open-interval-nonattainment · candidate
88. The Bounds of an Open Interval Are Not Its Maximum or Minimum
Use the entire nonempty bounded real open interval (12/13, 14/13), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
View answer and explanation
(12/13, 14/13, 14/13, 12/13, No, No, supremum_and_infimum_not_attained)
The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (12/13, 14/13, 14/13, 12/13, No, No, supremum_and_infimum_not_attained).
Reviewed step-by-step derivation
Supremum and infimum · qf-mod38-supinf-open-interval-nonattainment · candidate
89. The Bounds of an Open Interval Are Not Its Maximum or Minimum
Use the entire nonempty bounded real open interval (13/14, 15/14), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
View answer and explanation
(13/14, 15/14, 15/14, 13/14, No, No, supremum_and_infimum_not_attained)
The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (13/14, 15/14, 15/14, 13/14, No, No, supremum_and_infimum_not_attained).
Reviewed step-by-step derivation
Supremum and infimum · qf-mod38-supinf-open-interval-nonattainment · candidate
90. The Bounds of an Open Interval Are Not Its Maximum or Minimum
Use the entire nonempty bounded real open interval (14/15, 16/15), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
View answer and explanation
(14/15, 16/15, 16/15, 14/15, No, No, supremum_and_infimum_not_attained)
The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (14/15, 16/15, 16/15, 14/15, No, No, supremum_and_infimum_not_attained).
Reviewed step-by-step derivation
Supremum and infimum · qf-mod38-supinf-open-interval-nonattainment · candidate
91. The Bounds of an Open Interval Are Not Its Maximum or Minimum
Use the entire nonempty bounded real open interval (15/16, 17/16), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
View answer and explanation
(15/16, 17/16, 17/16, 15/16, No, No, supremum_and_infimum_not_attained)
The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (15/16, 17/16, 17/16, 15/16, No, No, supremum_and_infimum_not_attained).
Reviewed step-by-step derivation
Supremum and infimum · qf-mod38-supinf-open-interval-nonattainment · candidate
92. The Bounds of an Open Interval Are Not Its Maximum or Minimum
Use the entire nonempty bounded real open interval (16/17, 18/17), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
View answer and explanation
(16/17, 18/17, 18/17, 16/17, No, No, supremum_and_infimum_not_attained)
The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (16/17, 18/17, 18/17, 16/17, No, No, supremum_and_infimum_not_attained).
Reviewed step-by-step derivation
Supremum and infimum · qf-mod38-supinf-open-interval-nonattainment · candidate
93. The Bounds of an Open Interval Are Not Its Maximum or Minimum
Use the entire nonempty bounded real open interval (17/18, 19/18), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
View answer and explanation
(17/18, 19/18, 19/18, 17/18, No, No, supremum_and_infimum_not_attained)
The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (17/18, 19/18, 19/18, 17/18, No, No, supremum_and_infimum_not_attained).
Reviewed step-by-step derivation
Supremum and infimum · qf-mod38-supinf-open-interval-nonattainment · candidate
94. The Bounds of an Open Interval Are Not Its Maximum or Minimum
Use the entire nonempty bounded real open interval (18/19, 20/19), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
View answer and explanation
(18/19, 20/19, 20/19, 18/19, No, No, supremum_and_infimum_not_attained)
The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (18/19, 20/19, 20/19, 18/19, No, No, supremum_and_infimum_not_attained).
Reviewed step-by-step derivation
Supremum and infimum · qf-mod38-supinf-open-interval-nonattainment · candidate
95. The Bounds of an Open Interval Are Not Its Maximum or Minimum
Use the entire nonempty bounded real open interval (19/20, 21/20), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
View answer and explanation
(19/20, 21/20, 21/20, 19/20, No, No, supremum_and_infimum_not_attained)
The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (19/20, 21/20, 21/20, 19/20, No, No, supremum_and_infimum_not_attained).
Reviewed step-by-step derivation
Supremum and infimum · qf-mod38-supinf-open-interval-nonattainment · candidate
96. The Bounds of an Open Interval Are Not Its Maximum or Minimum
Use the entire nonempty bounded real open interval (20/21, 22/21), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
View answer and explanation
(20/21, 22/21, 22/21, 20/21, No, No, supremum_and_infimum_not_attained)
The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (20/21, 22/21, 22/21, 20/21, No, No, supremum_and_infimum_not_attained).
Reviewed step-by-step derivation
Supremum and infimum · qf-mod38-supinf-open-interval-nonattainment · candidate
97. The Bounds of an Open Interval Are Not Its Maximum or Minimum
Use the entire nonempty bounded real open interval (21/22, 23/22), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
View answer and explanation
(21/22, 23/22, 23/22, 21/22, No, No, supremum_and_infimum_not_attained)
The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (21/22, 23/22, 23/22, 21/22, No, No, supremum_and_infimum_not_attained).
Reviewed step-by-step derivation
Supremum and infimum · qf-mod38-supinf-open-interval-nonattainment · candidate
98. The Bounds of an Open Interval Are Not Its Maximum or Minimum
Use the entire nonempty bounded real open interval (22/23, 24/23), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
View answer and explanation
(22/23, 24/23, 24/23, 22/23, No, No, supremum_and_infimum_not_attained)
The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (22/23, 24/23, 24/23, 22/23, No, No, supremum_and_infimum_not_attained).
Reviewed step-by-step derivation
Supremum and infimum · qf-mod38-supinf-open-interval-nonattainment · candidate
99. The Bounds of an Open Interval Are Not Its Maximum or Minimum
Use the entire nonempty bounded real open interval (23/24, 25/24), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
View answer and explanation
(23/24, 25/24, 25/24, 23/24, No, No, supremum_and_infimum_not_attained)
The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (23/24, 25/24, 25/24, 23/24, No, No, supremum_and_infimum_not_attained).
Reviewed step-by-step derivation
Supremum and infimum · qf-mod38-supinf-open-interval-nonattainment · candidate
100. The Bounds of an Open Interval Are Not Its Maximum or Minimum
Use the entire nonempty bounded real open interval (24/25, 26/25), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
View answer and explanation
(24/25, 26/25, 26/25, 24/25, No, No, supremum_and_infimum_not_attained)
The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (24/25, 26/25, 26/25, 24/25, No, No, supremum_and_infimum_not_attained).
Reviewed step-by-step derivation
Supremum and infimum · qf-mod38-supinf-open-interval-nonattainment · candidate
Canonical question JSONL · English JSONL · Validation report