Common Strict Finite-Domain Witnesses, Supremum Gaps and Evidence Limits
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batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-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.
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Field-order and source-binding data
1. One Strict Witness Controls All Points of the Finite Domain
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[0, 1], coefficients=[1, -2], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/10. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).
View answer and explanation
(1/10, [[0, 9], [1, 19]], 19, Yes, same_N_for_all_finite_domain_points)
For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/10, [[0, 9], [1, 19]], 19, Yes, same_N_for_all_finite_domain_points).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-domain-witness · candidate
2. One Strict Witness Controls All Points of the Finite Domain
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[1, 2], coefficients=[2, -3], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/11. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).
View answer and explanation
(1/11, [[0, 21], [1, 32]], 32, Yes, same_N_for_all_finite_domain_points)
For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/11, [[0, 21], [1, 32]], 32, Yes, same_N_for_all_finite_domain_points).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-domain-witness · candidate
3. One Strict Witness Controls All Points of the Finite Domain
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[2, 3], coefficients=[3, -4], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/12. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).
View answer and explanation
(1/12, [[0, 35], [1, 47]], 47, Yes, same_N_for_all_finite_domain_points)
For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/12, [[0, 35], [1, 47]], 47, Yes, same_N_for_all_finite_domain_points).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-domain-witness · candidate
4. One Strict Witness Controls All Points of the Finite Domain
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[3, 4], coefficients=[4, -5], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/13. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).
View answer and explanation
(1/13, [[0, 51], [1, 64]], 64, Yes, same_N_for_all_finite_domain_points)
For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/13, [[0, 51], [1, 64]], 64, Yes, same_N_for_all_finite_domain_points).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-domain-witness · candidate
5. One Strict Witness Controls All Points of the Finite Domain
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[4, 5], coefficients=[5, -6], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/14. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).
View answer and explanation
(1/14, [[0, 69], [1, 83]], 83, Yes, same_N_for_all_finite_domain_points)
For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/14, [[0, 69], [1, 83]], 83, Yes, same_N_for_all_finite_domain_points).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-domain-witness · candidate
6. One Strict Witness Controls All Points of the Finite Domain
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[5, 6], coefficients=[6, -7], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/15. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).
View answer and explanation
(1/15, [[0, 89], [1, 104]], 104, Yes, same_N_for_all_finite_domain_points)
For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/15, [[0, 89], [1, 104]], 104, Yes, same_N_for_all_finite_domain_points).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-domain-witness · candidate
7. One Strict Witness Controls All Points of the Finite Domain
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[6, 7], coefficients=[7, -8], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/16. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).
View answer and explanation
(1/16, [[0, 111], [1, 127]], 127, Yes, same_N_for_all_finite_domain_points)
For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/16, [[0, 111], [1, 127]], 127, Yes, same_N_for_all_finite_domain_points).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-domain-witness · candidate
8. One Strict Witness Controls All Points of the Finite Domain
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[7, 8], coefficients=[8, -9], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/17. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).
View answer and explanation
(1/17, [[0, 135], [1, 152]], 152, Yes, same_N_for_all_finite_domain_points)
For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/17, [[0, 135], [1, 152]], 152, Yes, same_N_for_all_finite_domain_points).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-domain-witness · candidate
9. One Strict Witness Controls All Points of the Finite Domain
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[8, 9], coefficients=[9, -10], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/18. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).
View answer and explanation
(1/18, [[0, 161], [1, 179]], 179, Yes, same_N_for_all_finite_domain_points)
For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/18, [[0, 161], [1, 179]], 179, Yes, same_N_for_all_finite_domain_points).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-domain-witness · candidate
10. One Strict Witness Controls All Points of the Finite Domain
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[9, 10], coefficients=[10, -11], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/19. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).
View answer and explanation
(1/19, [[0, 189], [1, 208]], 208, Yes, same_N_for_all_finite_domain_points)
For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/19, [[0, 189], [1, 208]], 208, Yes, same_N_for_all_finite_domain_points).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-domain-witness · candidate
11. One Strict Witness Controls All Points of the Finite Domain
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[10, 11], coefficients=[11, -12], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/20. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).
View answer and explanation
(1/20, [[0, 219], [1, 239]], 239, Yes, same_N_for_all_finite_domain_points)
For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/20, [[0, 219], [1, 239]], 239, Yes, same_N_for_all_finite_domain_points).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-domain-witness · candidate
12. One Strict Witness Controls All Points of the Finite Domain
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[11, 12], coefficients=[12, -13], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/21. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).
View answer and explanation
(1/21, [[0, 251], [1, 272]], 272, Yes, same_N_for_all_finite_domain_points)
For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/21, [[0, 251], [1, 272]], 272, Yes, same_N_for_all_finite_domain_points).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-domain-witness · candidate
13. One Strict Witness Controls All Points of the Finite Domain
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[12, 13], coefficients=[13, -14], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/22. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).
View answer and explanation
(1/22, [[0, 285], [1, 307]], 307, Yes, same_N_for_all_finite_domain_points)
For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/22, [[0, 285], [1, 307]], 307, Yes, same_N_for_all_finite_domain_points).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-domain-witness · candidate
14. One Strict Witness Controls All Points of the Finite Domain
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[13, 14], coefficients=[14, -15], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/23. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).
View answer and explanation
(1/23, [[0, 321], [1, 344]], 344, Yes, same_N_for_all_finite_domain_points)
For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/23, [[0, 321], [1, 344]], 344, Yes, same_N_for_all_finite_domain_points).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-domain-witness · candidate
15. One Strict Witness Controls All Points of the Finite Domain
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[14, 15], coefficients=[15, -16], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/24. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).
View answer and explanation
(1/24, [[0, 359], [1, 383]], 383, Yes, same_N_for_all_finite_domain_points)
For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/24, [[0, 359], [1, 383]], 383, Yes, same_N_for_all_finite_domain_points).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-domain-witness · candidate
16. One Strict Witness Controls All Points of the Finite Domain
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[15, 16], coefficients=[16, -17], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/25. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).
View answer and explanation
(1/25, [[0, 399], [1, 424]], 424, Yes, same_N_for_all_finite_domain_points)
For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/25, [[0, 399], [1, 424]], 424, Yes, same_N_for_all_finite_domain_points).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-domain-witness · candidate
17. One Strict Witness Controls All Points of the Finite Domain
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[16, 17], coefficients=[17, -18], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/26. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).
View answer and explanation
(1/26, [[0, 441], [1, 467]], 467, Yes, same_N_for_all_finite_domain_points)
For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/26, [[0, 441], [1, 467]], 467, Yes, same_N_for_all_finite_domain_points).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-domain-witness · candidate
18. One Strict Witness Controls All Points of the Finite Domain
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[17, 18], coefficients=[18, -19], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/27. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).
View answer and explanation
(1/27, [[0, 485], [1, 512]], 512, Yes, same_N_for_all_finite_domain_points)
For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/27, [[0, 485], [1, 512]], 512, Yes, same_N_for_all_finite_domain_points).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-domain-witness · candidate
19. One Strict Witness Controls All Points of the Finite Domain
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[18, 19], coefficients=[19, -20], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/28. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).
View answer and explanation
(1/28, [[0, 531], [1, 559]], 559, Yes, same_N_for_all_finite_domain_points)
For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/28, [[0, 531], [1, 559]], 559, Yes, same_N_for_all_finite_domain_points).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-domain-witness · candidate
20. One Strict Witness Controls All Points of the Finite Domain
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[19, 20], coefficients=[20, -21], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/29. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).
View answer and explanation
(1/29, [[0, 579], [1, 608]], 608, Yes, same_N_for_all_finite_domain_points)
For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/29, [[0, 579], [1, 608]], 608, Yes, same_N_for_all_finite_domain_points).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-domain-witness · candidate
21. One Strict Witness Controls All Points of the Finite Domain
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[20, 21], coefficients=[21, -22], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/30. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).
View answer and explanation
(1/30, [[0, 629], [1, 659]], 659, Yes, same_N_for_all_finite_domain_points)
For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/30, [[0, 629], [1, 659]], 659, Yes, same_N_for_all_finite_domain_points).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-domain-witness · candidate
22. One Strict Witness Controls All Points of the Finite Domain
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[21, 22], coefficients=[22, -23], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/31. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).
View answer and explanation
(1/31, [[0, 681], [1, 712]], 712, Yes, same_N_for_all_finite_domain_points)
For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/31, [[0, 681], [1, 712]], 712, Yes, same_N_for_all_finite_domain_points).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-domain-witness · candidate
23. One Strict Witness Controls All Points of the Finite Domain
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[22, 23], coefficients=[23, -24], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/32. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).
View answer and explanation
(1/32, [[0, 735], [1, 767]], 767, Yes, same_N_for_all_finite_domain_points)
For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/32, [[0, 735], [1, 767]], 767, Yes, same_N_for_all_finite_domain_points).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-domain-witness · candidate
24. One Strict Witness Controls All Points of the Finite Domain
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[23, 24], coefficients=[24, -25], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/33. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).
View answer and explanation
(1/33, [[0, 791], [1, 824]], 824, Yes, same_N_for_all_finite_domain_points)
For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/33, [[0, 791], [1, 824]], 824, Yes, same_N_for_all_finite_domain_points).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-domain-witness · candidate
25. One Strict Witness Controls All Points of the Finite Domain
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[24, 25], coefficients=[25, -26], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/34. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate).
View answer and explanation
(1/34, [[0, 849], [1, 883]], 883, Yes, same_N_for_all_finite_domain_points)
For each point find the least nonnegative N_x with |coefficient[x]|/(N_x+shift)<epsilon, then take their finite maximum. Increasing denominators preserve the strict bound for every later n. The exact result, retaining the requested order, is (1/34, [[0, 849], [1, 883]], 883, Yes, same_N_for_all_finite_domain_points).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-domain-witness · candidate
26. Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[0, 1], coefficients=[1, -2], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=1. Give (n, sup_gap, certificate).
View answer and explanation
(1, 2/3, exact_finite_domain_supremum)
The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (1, 2/3, exact_finite_domain_supremum).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-sup-gap · candidate
27. Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[1, 2], coefficients=[2, -3], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=2. Give (n, sup_gap, certificate).
View answer and explanation
(2, 3/4, exact_finite_domain_supremum)
The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (2, 3/4, exact_finite_domain_supremum).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-sup-gap · candidate
28. Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[2, 3], coefficients=[3, -4], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=3. Give (n, sup_gap, certificate).
View answer and explanation
(3, 4/5, exact_finite_domain_supremum)
The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (3, 4/5, exact_finite_domain_supremum).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-sup-gap · candidate
29. Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[3, 4], coefficients=[4, -5], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=4. Give (n, sup_gap, certificate).
View answer and explanation
(4, 5/6, exact_finite_domain_supremum)
The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (4, 5/6, exact_finite_domain_supremum).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-sup-gap · candidate
30. Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[4, 5], coefficients=[5, -6], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=5. Give (n, sup_gap, certificate).
View answer and explanation
(5, 6/7, exact_finite_domain_supremum)
The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (5, 6/7, exact_finite_domain_supremum).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-sup-gap · candidate
31. Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[5, 6], coefficients=[6, -7], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=6. Give (n, sup_gap, certificate).
View answer and explanation
(6, 7/8, exact_finite_domain_supremum)
The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (6, 7/8, exact_finite_domain_supremum).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-sup-gap · candidate
32. Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[6, 7], coefficients=[7, -8], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=7. Give (n, sup_gap, certificate).
View answer and explanation
(7, 8/9, exact_finite_domain_supremum)
The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (7, 8/9, exact_finite_domain_supremum).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-sup-gap · candidate
33. Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[7, 8], coefficients=[8, -9], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=8. Give (n, sup_gap, certificate).
View answer and explanation
(8, 9/10, exact_finite_domain_supremum)
The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (8, 9/10, exact_finite_domain_supremum).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-sup-gap · candidate
34. Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[8, 9], coefficients=[9, -10], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=9. Give (n, sup_gap, certificate).
View answer and explanation
(9, 10/11, exact_finite_domain_supremum)
The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (9, 10/11, exact_finite_domain_supremum).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-sup-gap · candidate
35. Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[9, 10], coefficients=[10, -11], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=10. Give (n, sup_gap, certificate).
View answer and explanation
(10, 11/12, exact_finite_domain_supremum)
The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (10, 11/12, exact_finite_domain_supremum).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-sup-gap · candidate
36. Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[10, 11], coefficients=[11, -12], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=11. Give (n, sup_gap, certificate).
View answer and explanation
(11, 12/13, exact_finite_domain_supremum)
The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (11, 12/13, exact_finite_domain_supremum).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-sup-gap · candidate
37. Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[11, 12], coefficients=[12, -13], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=12. Give (n, sup_gap, certificate).
View answer and explanation
(12, 13/14, exact_finite_domain_supremum)
The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (12, 13/14, exact_finite_domain_supremum).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-sup-gap · candidate
38. Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[12, 13], coefficients=[13, -14], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=13. Give (n, sup_gap, certificate).
View answer and explanation
(13, 14/15, exact_finite_domain_supremum)
The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (13, 14/15, exact_finite_domain_supremum).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-sup-gap · candidate
39. Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[13, 14], coefficients=[14, -15], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=14. Give (n, sup_gap, certificate).
View answer and explanation
(14, 15/16, exact_finite_domain_supremum)
The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (14, 15/16, exact_finite_domain_supremum).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-sup-gap · candidate
40. Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[14, 15], coefficients=[15, -16], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=15. Give (n, sup_gap, certificate).
View answer and explanation
(15, 16/17, exact_finite_domain_supremum)
The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (15, 16/17, exact_finite_domain_supremum).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-sup-gap · candidate
41. Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[15, 16], coefficients=[16, -17], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=16. Give (n, sup_gap, certificate).
View answer and explanation
(16, 17/18, exact_finite_domain_supremum)
The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (16, 17/18, exact_finite_domain_supremum).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-sup-gap · candidate
42. Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[16, 17], coefficients=[17, -18], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=17. Give (n, sup_gap, certificate).
View answer and explanation
(17, 18/19, exact_finite_domain_supremum)
The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (17, 18/19, exact_finite_domain_supremum).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-sup-gap · candidate
43. Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[17, 18], coefficients=[18, -19], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=18. Give (n, sup_gap, certificate).
View answer and explanation
(18, 19/20, exact_finite_domain_supremum)
The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (18, 19/20, exact_finite_domain_supremum).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-sup-gap · candidate
44. Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[18, 19], coefficients=[19, -20], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=19. Give (n, sup_gap, certificate).
View answer and explanation
(19, 20/21, exact_finite_domain_supremum)
The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (19, 20/21, exact_finite_domain_supremum).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-sup-gap · candidate
45. Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[19, 20], coefficients=[20, -21], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=20. Give (n, sup_gap, certificate).
View answer and explanation
(20, 21/22, exact_finite_domain_supremum)
The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (20, 21/22, exact_finite_domain_supremum).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-sup-gap · candidate
46. Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[20, 21], coefficients=[21, -22], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=21. Give (n, sup_gap, certificate).
View answer and explanation
(21, 22/23, exact_finite_domain_supremum)
The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (21, 22/23, exact_finite_domain_supremum).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-sup-gap · candidate
47. Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[21, 22], coefficients=[22, -23], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=22. Give (n, sup_gap, certificate).
View answer and explanation
(22, 23/24, exact_finite_domain_supremum)
The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (22, 23/24, exact_finite_domain_supremum).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-sup-gap · candidate
48. Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[22, 23], coefficients=[23, -24], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=23. Give (n, sup_gap, certificate).
View answer and explanation
(23, 24/25, exact_finite_domain_supremum)
The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (23, 24/25, exact_finite_domain_supremum).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-sup-gap · candidate
49. Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[23, 24], coefficients=[24, -25], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=24. Give (n, sup_gap, certificate).
View answer and explanation
(24, 25/26, exact_finite_domain_supremum)
The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (24, 25/26, exact_finite_domain_supremum).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-sup-gap · candidate
50. Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[24, 25], coefficients=[25, -26], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=25. Give (n, sup_gap, certificate).
View answer and explanation
(25, 26/27, exact_finite_domain_supremum)
The exact supremum is max_x |coefficient[x]|/(n+shift). Retain the source direct-ratio text, which may be unreduced, instead of changing its numerical bytes during translation. The exact result, retaining the requested order, is (25, 26/27, exact_finite_domain_supremum).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-sup-gap · candidate
51. A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[0, 1], coefficients=[1, -2], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/10. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).
View answer and explanation
([[0, 9], [1, 19]], 19, Yes, finite_maximum_only_not_infinite_domain_implication)
List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 9], [1, 19]], 19, Yes, finite_maximum_only_not_infinite_domain_implication).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-pointwise-max · candidate
52. A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[1, 2], coefficients=[2, -3], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/11. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).
View answer and explanation
([[0, 21], [1, 32]], 32, Yes, finite_maximum_only_not_infinite_domain_implication)
List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 21], [1, 32]], 32, Yes, finite_maximum_only_not_infinite_domain_implication).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-pointwise-max · candidate
53. A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[2, 3], coefficients=[3, -4], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/12. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).
View answer and explanation
([[0, 35], [1, 47]], 47, Yes, finite_maximum_only_not_infinite_domain_implication)
List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 35], [1, 47]], 47, Yes, finite_maximum_only_not_infinite_domain_implication).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-pointwise-max · candidate
54. A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[3, 4], coefficients=[4, -5], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/13. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).
View answer and explanation
([[0, 51], [1, 64]], 64, Yes, finite_maximum_only_not_infinite_domain_implication)
List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 51], [1, 64]], 64, Yes, finite_maximum_only_not_infinite_domain_implication).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-pointwise-max · candidate
55. A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[4, 5], coefficients=[5, -6], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/14. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).
View answer and explanation
([[0, 69], [1, 83]], 83, Yes, finite_maximum_only_not_infinite_domain_implication)
List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 69], [1, 83]], 83, Yes, finite_maximum_only_not_infinite_domain_implication).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-pointwise-max · candidate
56. A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[5, 6], coefficients=[6, -7], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/15. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).
View answer and explanation
([[0, 89], [1, 104]], 104, Yes, finite_maximum_only_not_infinite_domain_implication)
List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 89], [1, 104]], 104, Yes, finite_maximum_only_not_infinite_domain_implication).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-pointwise-max · candidate
57. A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[6, 7], coefficients=[7, -8], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/16. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).
View answer and explanation
([[0, 111], [1, 127]], 127, Yes, finite_maximum_only_not_infinite_domain_implication)
List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 111], [1, 127]], 127, Yes, finite_maximum_only_not_infinite_domain_implication).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-pointwise-max · candidate
58. A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[7, 8], coefficients=[8, -9], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/17. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).
View answer and explanation
([[0, 135], [1, 152]], 152, Yes, finite_maximum_only_not_infinite_domain_implication)
List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 135], [1, 152]], 152, Yes, finite_maximum_only_not_infinite_domain_implication).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-pointwise-max · candidate
59. A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[8, 9], coefficients=[9, -10], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/18. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).
View answer and explanation
([[0, 161], [1, 179]], 179, Yes, finite_maximum_only_not_infinite_domain_implication)
List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 161], [1, 179]], 179, Yes, finite_maximum_only_not_infinite_domain_implication).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-pointwise-max · candidate
60. A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[9, 10], coefficients=[10, -11], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/19. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).
View answer and explanation
([[0, 189], [1, 208]], 208, Yes, finite_maximum_only_not_infinite_domain_implication)
List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 189], [1, 208]], 208, Yes, finite_maximum_only_not_infinite_domain_implication).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-pointwise-max · candidate
61. A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[10, 11], coefficients=[11, -12], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/20. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).
View answer and explanation
([[0, 219], [1, 239]], 239, Yes, finite_maximum_only_not_infinite_domain_implication)
List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 219], [1, 239]], 239, Yes, finite_maximum_only_not_infinite_domain_implication).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-pointwise-max · candidate
62. A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[11, 12], coefficients=[12, -13], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/21. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).
View answer and explanation
([[0, 251], [1, 272]], 272, Yes, finite_maximum_only_not_infinite_domain_implication)
List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 251], [1, 272]], 272, Yes, finite_maximum_only_not_infinite_domain_implication).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-pointwise-max · candidate
63. A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[12, 13], coefficients=[13, -14], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/22. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).
View answer and explanation
([[0, 285], [1, 307]], 307, Yes, finite_maximum_only_not_infinite_domain_implication)
List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 285], [1, 307]], 307, Yes, finite_maximum_only_not_infinite_domain_implication).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-pointwise-max · candidate
64. A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[13, 14], coefficients=[14, -15], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/23. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).
View answer and explanation
([[0, 321], [1, 344]], 344, Yes, finite_maximum_only_not_infinite_domain_implication)
List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 321], [1, 344]], 344, Yes, finite_maximum_only_not_infinite_domain_implication).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-pointwise-max · candidate
65. A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[14, 15], coefficients=[15, -16], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/24. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).
View answer and explanation
([[0, 359], [1, 383]], 383, Yes, finite_maximum_only_not_infinite_domain_implication)
List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 359], [1, 383]], 383, Yes, finite_maximum_only_not_infinite_domain_implication).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-pointwise-max · candidate
66. A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[15, 16], coefficients=[16, -17], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/25. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).
View answer and explanation
([[0, 399], [1, 424]], 424, Yes, finite_maximum_only_not_infinite_domain_implication)
List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 399], [1, 424]], 424, Yes, finite_maximum_only_not_infinite_domain_implication).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-pointwise-max · candidate
67. A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[16, 17], coefficients=[17, -18], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/26. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).
View answer and explanation
([[0, 441], [1, 467]], 467, Yes, finite_maximum_only_not_infinite_domain_implication)
List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 441], [1, 467]], 467, Yes, finite_maximum_only_not_infinite_domain_implication).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-pointwise-max · candidate
68. A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[17, 18], coefficients=[18, -19], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/27. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).
View answer and explanation
([[0, 485], [1, 512]], 512, Yes, finite_maximum_only_not_infinite_domain_implication)
List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 485], [1, 512]], 512, Yes, finite_maximum_only_not_infinite_domain_implication).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-pointwise-max · candidate
69. A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[18, 19], coefficients=[19, -20], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/28. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).
View answer and explanation
([[0, 531], [1, 559]], 559, Yes, finite_maximum_only_not_infinite_domain_implication)
List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 531], [1, 559]], 559, Yes, finite_maximum_only_not_infinite_domain_implication).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-pointwise-max · candidate
70. A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[19, 20], coefficients=[20, -21], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/29. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).
View answer and explanation
([[0, 579], [1, 608]], 608, Yes, finite_maximum_only_not_infinite_domain_implication)
List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 579], [1, 608]], 608, Yes, finite_maximum_only_not_infinite_domain_implication).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-pointwise-max · candidate
71. A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[20, 21], coefficients=[21, -22], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/30. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).
View answer and explanation
([[0, 629], [1, 659]], 659, Yes, finite_maximum_only_not_infinite_domain_implication)
List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 629], [1, 659]], 659, Yes, finite_maximum_only_not_infinite_domain_implication).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-pointwise-max · candidate
72. A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[21, 22], coefficients=[22, -23], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/31. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).
View answer and explanation
([[0, 681], [1, 712]], 712, Yes, finite_maximum_only_not_infinite_domain_implication)
List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 681], [1, 712]], 712, Yes, finite_maximum_only_not_infinite_domain_implication).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-pointwise-max · candidate
73. A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[22, 23], coefficients=[23, -24], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/32. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).
View answer and explanation
([[0, 735], [1, 767]], 767, Yes, finite_maximum_only_not_infinite_domain_implication)
List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 735], [1, 767]], 767, Yes, finite_maximum_only_not_infinite_domain_implication).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-pointwise-max · candidate
74. A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[23, 24], coefficients=[24, -25], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/33. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).
View answer and explanation
([[0, 791], [1, 824]], 824, Yes, finite_maximum_only_not_infinite_domain_implication)
List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 791], [1, 824]], 824, Yes, finite_maximum_only_not_infinite_domain_implication).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-pointwise-max · candidate
75. A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[24, 25], coefficients=[25, -26], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/34. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).
View answer and explanation
([[0, 849], [1, 883]], 883, Yes, finite_maximum_only_not_infinite_domain_implication)
List pointwise strict witnesses in domain order and take their maximum. That supplies a common N on this complete finite domain only, not a general implication from pointwise to uniform convergence. The exact result, retaining the requested order, is ([[0, 849], [1, 883]], 883, Yes, finite_maximum_only_not_infinite_domain_implication).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-pointwise-max · candidate
76. A Finite Prefix Does Not Supply the Uniform All-n Obligation
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[0, 1], coefficients=[1, -2], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=1. Give (prefix_length, sufficient, boundary).
View answer and explanation
(1, No, finite_prefix_not_uniform_convergence_certificate)
The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (1, No, finite_prefix_not_uniform_convergence_certificate).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-prefix-boundary · candidate
77. A Finite Prefix Does Not Supply the Uniform All-n Obligation
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[1, 2], coefficients=[2, -3], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=2. Give (prefix_length, sufficient, boundary).
View answer and explanation
(2, No, finite_prefix_not_uniform_convergence_certificate)
The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (2, No, finite_prefix_not_uniform_convergence_certificate).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-prefix-boundary · candidate
78. A Finite Prefix Does Not Supply the Uniform All-n Obligation
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[2, 3], coefficients=[3, -4], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=3. Give (prefix_length, sufficient, boundary).
View answer and explanation
(3, No, finite_prefix_not_uniform_convergence_certificate)
The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (3, No, finite_prefix_not_uniform_convergence_certificate).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-prefix-boundary · candidate
79. A Finite Prefix Does Not Supply the Uniform All-n Obligation
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[3, 4], coefficients=[4, -5], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=4. Give (prefix_length, sufficient, boundary).
View answer and explanation
(4, No, finite_prefix_not_uniform_convergence_certificate)
The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (4, No, finite_prefix_not_uniform_convergence_certificate).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-prefix-boundary · candidate
80. A Finite Prefix Does Not Supply the Uniform All-n Obligation
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[4, 5], coefficients=[5, -6], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=5. Give (prefix_length, sufficient, boundary).
View answer and explanation
(5, No, finite_prefix_not_uniform_convergence_certificate)
The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (5, No, finite_prefix_not_uniform_convergence_certificate).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-prefix-boundary · candidate
81. A Finite Prefix Does Not Supply the Uniform All-n Obligation
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[5, 6], coefficients=[6, -7], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=6. Give (prefix_length, sufficient, boundary).
View answer and explanation
(6, No, finite_prefix_not_uniform_convergence_certificate)
The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (6, No, finite_prefix_not_uniform_convergence_certificate).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-prefix-boundary · candidate
82. A Finite Prefix Does Not Supply the Uniform All-n Obligation
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[6, 7], coefficients=[7, -8], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=7. Give (prefix_length, sufficient, boundary).
View answer and explanation
(7, No, finite_prefix_not_uniform_convergence_certificate)
The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (7, No, finite_prefix_not_uniform_convergence_certificate).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-prefix-boundary · candidate
83. A Finite Prefix Does Not Supply the Uniform All-n Obligation
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[7, 8], coefficients=[8, -9], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=8. Give (prefix_length, sufficient, boundary).
View answer and explanation
(8, No, finite_prefix_not_uniform_convergence_certificate)
The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (8, No, finite_prefix_not_uniform_convergence_certificate).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-prefix-boundary · candidate
84. A Finite Prefix Does Not Supply the Uniform All-n Obligation
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[8, 9], coefficients=[9, -10], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=9. Give (prefix_length, sufficient, boundary).
View answer and explanation
(9, No, finite_prefix_not_uniform_convergence_certificate)
The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (9, No, finite_prefix_not_uniform_convergence_certificate).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-prefix-boundary · candidate
85. A Finite Prefix Does Not Supply the Uniform All-n Obligation
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[9, 10], coefficients=[10, -11], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=10. Give (prefix_length, sufficient, boundary).
View answer and explanation
(10, No, finite_prefix_not_uniform_convergence_certificate)
The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (10, No, finite_prefix_not_uniform_convergence_certificate).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-prefix-boundary · candidate
86. A Finite Prefix Does Not Supply the Uniform All-n Obligation
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[10, 11], coefficients=[11, -12], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=11. Give (prefix_length, sufficient, boundary).
View answer and explanation
(11, No, finite_prefix_not_uniform_convergence_certificate)
The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (11, No, finite_prefix_not_uniform_convergence_certificate).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-prefix-boundary · candidate
87. A Finite Prefix Does Not Supply the Uniform All-n Obligation
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[11, 12], coefficients=[12, -13], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=12. Give (prefix_length, sufficient, boundary).
View answer and explanation
(12, No, finite_prefix_not_uniform_convergence_certificate)
The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (12, No, finite_prefix_not_uniform_convergence_certificate).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-prefix-boundary · candidate
88. A Finite Prefix Does Not Supply the Uniform All-n Obligation
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[12, 13], coefficients=[13, -14], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=13. Give (prefix_length, sufficient, boundary).
View answer and explanation
(13, No, finite_prefix_not_uniform_convergence_certificate)
The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (13, No, finite_prefix_not_uniform_convergence_certificate).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-prefix-boundary · candidate
89. A Finite Prefix Does Not Supply the Uniform All-n Obligation
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[13, 14], coefficients=[14, -15], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=14. Give (prefix_length, sufficient, boundary).
View answer and explanation
(14, No, finite_prefix_not_uniform_convergence_certificate)
The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (14, No, finite_prefix_not_uniform_convergence_certificate).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-prefix-boundary · candidate
90. A Finite Prefix Does Not Supply the Uniform All-n Obligation
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[14, 15], coefficients=[15, -16], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=15. Give (prefix_length, sufficient, boundary).
View answer and explanation
(15, No, finite_prefix_not_uniform_convergence_certificate)
The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (15, No, finite_prefix_not_uniform_convergence_certificate).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-prefix-boundary · candidate
91. A Finite Prefix Does Not Supply the Uniform All-n Obligation
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[15, 16], coefficients=[16, -17], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=16. Give (prefix_length, sufficient, boundary).
View answer and explanation
(16, No, finite_prefix_not_uniform_convergence_certificate)
The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (16, No, finite_prefix_not_uniform_convergence_certificate).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-prefix-boundary · candidate
92. A Finite Prefix Does Not Supply the Uniform All-n Obligation
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[16, 17], coefficients=[17, -18], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=17. Give (prefix_length, sufficient, boundary).
View answer and explanation
(17, No, finite_prefix_not_uniform_convergence_certificate)
The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (17, No, finite_prefix_not_uniform_convergence_certificate).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-prefix-boundary · candidate
93. A Finite Prefix Does Not Supply the Uniform All-n Obligation
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[17, 18], coefficients=[18, -19], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=18. Give (prefix_length, sufficient, boundary).
View answer and explanation
(18, No, finite_prefix_not_uniform_convergence_certificate)
The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (18, No, finite_prefix_not_uniform_convergence_certificate).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-prefix-boundary · candidate
94. A Finite Prefix Does Not Supply the Uniform All-n Obligation
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[18, 19], coefficients=[19, -20], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=19. Give (prefix_length, sufficient, boundary).
View answer and explanation
(19, No, finite_prefix_not_uniform_convergence_certificate)
The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (19, No, finite_prefix_not_uniform_convergence_certificate).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-prefix-boundary · candidate
95. A Finite Prefix Does Not Supply the Uniform All-n Obligation
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[19, 20], coefficients=[20, -21], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=20. Give (prefix_length, sufficient, boundary).
View answer and explanation
(20, No, finite_prefix_not_uniform_convergence_certificate)
The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (20, No, finite_prefix_not_uniform_convergence_certificate).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-prefix-boundary · candidate
96. A Finite Prefix Does Not Supply the Uniform All-n Obligation
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[20, 21], coefficients=[21, -22], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=21. Give (prefix_length, sufficient, boundary).
View answer and explanation
(21, No, finite_prefix_not_uniform_convergence_certificate)
The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (21, No, finite_prefix_not_uniform_convergence_certificate).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-prefix-boundary · candidate
97. A Finite Prefix Does Not Supply the Uniform All-n Obligation
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[21, 22], coefficients=[22, -23], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=22. Give (prefix_length, sufficient, boundary).
View answer and explanation
(22, No, finite_prefix_not_uniform_convergence_certificate)
The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (22, No, finite_prefix_not_uniform_convergence_certificate).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-prefix-boundary · candidate
98. A Finite Prefix Does Not Supply the Uniform All-n Obligation
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[22, 23], coefficients=[23, -24], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=23. Give (prefix_length, sufficient, boundary).
View answer and explanation
(23, No, finite_prefix_not_uniform_convergence_certificate)
The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (23, No, finite_prefix_not_uniform_convergence_certificate).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-prefix-boundary · candidate
99. A Finite Prefix Does Not Supply the Uniform All-n Obligation
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[23, 24], coefficients=[24, -25], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=24. Give (prefix_length, sufficient, boundary).
View answer and explanation
(24, No, finite_prefix_not_uniform_convergence_certificate)
The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (24, No, finite_prefix_not_uniform_convergence_certificate).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-prefix-boundary · candidate
100. A Finite Prefix Does Not Supply the Uniform All-n Obligation
Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[24, 25], coefficients=[25, -26], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Observed prefix_length=25. Give (prefix_length, sufficient, boundary).
View answer and explanation
(25, No, finite_prefix_not_uniform_convergence_certificate)
The No concerns sufficiency of finite-prefix evidence, not failure of the separately supplied symbolic family. Uniform convergence needs one witness controlling all points and every later n. The exact result, retaining the requested order, is (25, No, finite_prefix_not_uniform_convergence_certificate).
Reviewed step-by-step derivation
Uniform convergence of functions · qf-mod35-uniform-finite-prefix-boundary · candidate
Canonical question JSONL · English JSONL · Validation report