# 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.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

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## 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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0100/)

Uniform convergence of functions · qf-mod35-uniform-finite-prefix-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations/validation-report.json)