Signed Pointwise Values, Strict Epsilon Tails and Finite-Domain Maxima
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batch-20261014-advance-20261103-r02-pointwise-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.
1. Keep the Signed Function Error Separate from Its Absolute Bound
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[0, 1], coefficients=[1, -2], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=1, point=0. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
View answer and explanation
(1, 0, 1/3, 0, 1/3)
Evaluate the exact rational tail at the selected point. A negative coefficient gives a negative signed error; do not replace that field with its absolute value. One value is not the full pointwise-limit proof. The exact result, retaining the requested order, is (1, 0, 1/3, 0, 1/3).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-evaluate · candidate
2. Keep the Signed Function Error Separate from Its Absolute Bound
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[1, 2], coefficients=[2, -3], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=2, point=1. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
View answer and explanation
(2, 1, 5/4, 2, -3/4)
Evaluate the exact rational tail at the selected point. A negative coefficient gives a negative signed error; do not replace that field with its absolute value. One value is not the full pointwise-limit proof. The exact result, retaining the requested order, is (2, 1, 5/4, 2, -3/4).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-evaluate · candidate
3. Keep the Signed Function Error Separate from Its Absolute Bound
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[2, 3], coefficients=[3, -4], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=3, point=0. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
View answer and explanation
(3, 0, 13/5, 2, 3/5)
Evaluate the exact rational tail at the selected point. A negative coefficient gives a negative signed error; do not replace that field with its absolute value. One value is not the full pointwise-limit proof. The exact result, retaining the requested order, is (3, 0, 13/5, 2, 3/5).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-evaluate · candidate
4. Keep the Signed Function Error Separate from Its Absolute Bound
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[3, 4], coefficients=[4, -5], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=4, point=1. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
View answer and explanation
(4, 1, 19/6, 4, -5/6)
Evaluate the exact rational tail at the selected point. A negative coefficient gives a negative signed error; do not replace that field with its absolute value. One value is not the full pointwise-limit proof. The exact result, retaining the requested order, is (4, 1, 19/6, 4, -5/6).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-evaluate · candidate
5. Keep the Signed Function Error Separate from Its Absolute Bound
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[4, 5], coefficients=[5, -6], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=5, point=0. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
View answer and explanation
(5, 0, 33/7, 4, 5/7)
Evaluate the exact rational tail at the selected point. A negative coefficient gives a negative signed error; do not replace that field with its absolute value. One value is not the full pointwise-limit proof. The exact result, retaining the requested order, is (5, 0, 33/7, 4, 5/7).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-evaluate · candidate
6. Keep the Signed Function Error Separate from Its Absolute Bound
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[5, 6], coefficients=[6, -7], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=6, point=1. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
View answer and explanation
(6, 1, 41/8, 6, -7/8)
Evaluate the exact rational tail at the selected point. A negative coefficient gives a negative signed error; do not replace that field with its absolute value. One value is not the full pointwise-limit proof. The exact result, retaining the requested order, is (6, 1, 41/8, 6, -7/8).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-evaluate · candidate
7. Keep the Signed Function Error Separate from Its Absolute Bound
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[6, 7], coefficients=[7, -8], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=7, point=0. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
View answer and explanation
(7, 0, 61/9, 6, 7/9)
Evaluate the exact rational tail at the selected point. A negative coefficient gives a negative signed error; do not replace that field with its absolute value. One value is not the full pointwise-limit proof. The exact result, retaining the requested order, is (7, 0, 61/9, 6, 7/9).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-evaluate · candidate
8. Keep the Signed Function Error Separate from Its Absolute Bound
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[7, 8], coefficients=[8, -9], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=8, point=1. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
View answer and explanation
(8, 1, 71/10, 8, -9/10)
Evaluate the exact rational tail at the selected point. A negative coefficient gives a negative signed error; do not replace that field with its absolute value. One value is not the full pointwise-limit proof. The exact result, retaining the requested order, is (8, 1, 71/10, 8, -9/10).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-evaluate · candidate
9. Keep the Signed Function Error Separate from Its Absolute Bound
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[8, 9], coefficients=[9, -10], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=9, point=0. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
View answer and explanation
(9, 0, 97/11, 8, 9/11)
Evaluate the exact rational tail at the selected point. A negative coefficient gives a negative signed error; do not replace that field with its absolute value. One value is not the full pointwise-limit proof. The exact result, retaining the requested order, is (9, 0, 97/11, 8, 9/11).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-evaluate · candidate
10. Keep the Signed Function Error Separate from Its Absolute Bound
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[9, 10], coefficients=[10, -11], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=10, point=1. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
View answer and explanation
(10, 1, 109/12, 10, -11/12)
Evaluate the exact rational tail at the selected point. A negative coefficient gives a negative signed error; do not replace that field with its absolute value. One value is not the full pointwise-limit proof. The exact result, retaining the requested order, is (10, 1, 109/12, 10, -11/12).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-evaluate · candidate
11. Keep the Signed Function Error Separate from Its Absolute Bound
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[10, 11], coefficients=[11, -12], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=11, point=0. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
View answer and explanation
(11, 0, 141/13, 10, 11/13)
Evaluate the exact rational tail at the selected point. A negative coefficient gives a negative signed error; do not replace that field with its absolute value. One value is not the full pointwise-limit proof. The exact result, retaining the requested order, is (11, 0, 141/13, 10, 11/13).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-evaluate · candidate
12. Keep the Signed Function Error Separate from Its Absolute Bound
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[11, 12], coefficients=[12, -13], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=12, point=1. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
View answer and explanation
(12, 1, 155/14, 12, -13/14)
Evaluate the exact rational tail at the selected point. A negative coefficient gives a negative signed error; do not replace that field with its absolute value. One value is not the full pointwise-limit proof. The exact result, retaining the requested order, is (12, 1, 155/14, 12, -13/14).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-evaluate · candidate
13. Keep the Signed Function Error Separate from Its Absolute Bound
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[12, 13], coefficients=[13, -14], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=13, point=0. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
View answer and explanation
(13, 0, 193/15, 12, 13/15)
Evaluate the exact rational tail at the selected point. A negative coefficient gives a negative signed error; do not replace that field with its absolute value. One value is not the full pointwise-limit proof. The exact result, retaining the requested order, is (13, 0, 193/15, 12, 13/15).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-evaluate · candidate
14. Keep the Signed Function Error Separate from Its Absolute Bound
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[13, 14], coefficients=[14, -15], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=14, point=1. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
View answer and explanation
(14, 1, 209/16, 14, -15/16)
Evaluate the exact rational tail at the selected point. A negative coefficient gives a negative signed error; do not replace that field with its absolute value. One value is not the full pointwise-limit proof. The exact result, retaining the requested order, is (14, 1, 209/16, 14, -15/16).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-evaluate · candidate
15. Keep the Signed Function Error Separate from Its Absolute Bound
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[14, 15], coefficients=[15, -16], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=15, point=0. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
View answer and explanation
(15, 0, 253/17, 14, 15/17)
Evaluate the exact rational tail at the selected point. A negative coefficient gives a negative signed error; do not replace that field with its absolute value. One value is not the full pointwise-limit proof. The exact result, retaining the requested order, is (15, 0, 253/17, 14, 15/17).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-evaluate · candidate
16. Keep the Signed Function Error Separate from Its Absolute Bound
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[15, 16], coefficients=[16, -17], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=16, point=1. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
View answer and explanation
(16, 1, 271/18, 16, -17/18)
Evaluate the exact rational tail at the selected point. A negative coefficient gives a negative signed error; do not replace that field with its absolute value. One value is not the full pointwise-limit proof. The exact result, retaining the requested order, is (16, 1, 271/18, 16, -17/18).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-evaluate · candidate
17. Keep the Signed Function Error Separate from Its Absolute Bound
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[16, 17], coefficients=[17, -18], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=17, point=0. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
View answer and explanation
(17, 0, 321/19, 16, 17/19)
Evaluate the exact rational tail at the selected point. A negative coefficient gives a negative signed error; do not replace that field with its absolute value. One value is not the full pointwise-limit proof. The exact result, retaining the requested order, is (17, 0, 321/19, 16, 17/19).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-evaluate · candidate
18. Keep the Signed Function Error Separate from Its Absolute Bound
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[17, 18], coefficients=[18, -19], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=18, point=1. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
View answer and explanation
(18, 1, 341/20, 18, -19/20)
Evaluate the exact rational tail at the selected point. A negative coefficient gives a negative signed error; do not replace that field with its absolute value. One value is not the full pointwise-limit proof. The exact result, retaining the requested order, is (18, 1, 341/20, 18, -19/20).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-evaluate · candidate
19. Keep the Signed Function Error Separate from Its Absolute Bound
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[18, 19], coefficients=[19, -20], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=19, point=0. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
View answer and explanation
(19, 0, 397/21, 18, 19/21)
Evaluate the exact rational tail at the selected point. A negative coefficient gives a negative signed error; do not replace that field with its absolute value. One value is not the full pointwise-limit proof. The exact result, retaining the requested order, is (19, 0, 397/21, 18, 19/21).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-evaluate · candidate
20. Keep the Signed Function Error Separate from Its Absolute Bound
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[19, 20], coefficients=[20, -21], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=20, point=1. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
View answer and explanation
(20, 1, 419/22, 20, -21/22)
Evaluate the exact rational tail at the selected point. A negative coefficient gives a negative signed error; do not replace that field with its absolute value. One value is not the full pointwise-limit proof. The exact result, retaining the requested order, is (20, 1, 419/22, 20, -21/22).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-evaluate · candidate
21. Keep the Signed Function Error Separate from Its Absolute Bound
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[20, 21], coefficients=[21, -22], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=21, point=0. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
View answer and explanation
(21, 0, 481/23, 20, 21/23)
Evaluate the exact rational tail at the selected point. A negative coefficient gives a negative signed error; do not replace that field with its absolute value. One value is not the full pointwise-limit proof. The exact result, retaining the requested order, is (21, 0, 481/23, 20, 21/23).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-evaluate · candidate
22. Keep the Signed Function Error Separate from Its Absolute Bound
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[21, 22], coefficients=[22, -23], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=22, point=1. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
View answer and explanation
(22, 1, 505/24, 22, -23/24)
Evaluate the exact rational tail at the selected point. A negative coefficient gives a negative signed error; do not replace that field with its absolute value. One value is not the full pointwise-limit proof. The exact result, retaining the requested order, is (22, 1, 505/24, 22, -23/24).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-evaluate · candidate
23. Keep the Signed Function Error Separate from Its Absolute Bound
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[22, 23], coefficients=[23, -24], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=23, point=0. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
View answer and explanation
(23, 0, 573/25, 22, 23/25)
Evaluate the exact rational tail at the selected point. A negative coefficient gives a negative signed error; do not replace that field with its absolute value. One value is not the full pointwise-limit proof. The exact result, retaining the requested order, is (23, 0, 573/25, 22, 23/25).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-evaluate · candidate
24. Keep the Signed Function Error Separate from Its Absolute Bound
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[23, 24], coefficients=[24, -25], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=24, point=1. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
View answer and explanation
(24, 1, 599/26, 24, -25/26)
Evaluate the exact rational tail at the selected point. A negative coefficient gives a negative signed error; do not replace that field with its absolute value. One value is not the full pointwise-limit proof. The exact result, retaining the requested order, is (24, 1, 599/26, 24, -25/26).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-evaluate · candidate
25. Keep the Signed Function Error Separate from Its Absolute Bound
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[24, 25], coefficients=[25, -26], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. n=25, point=0. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed.
View answer and explanation
(25, 0, 673/27, 24, 25/27)
Evaluate the exact rational tail at the selected point. A negative coefficient gives a negative signed error; do not replace that field with its absolute value. One value is not the full pointwise-limit proof. The exact result, retaining the requested order, is (25, 0, 673/27, 24, 25/27).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-evaluate · candidate
26. A Strict Pointwise Tail Witness Can Depend on the Point
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[0, 1], coefficients=[1, -2], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/10, point=0. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
View answer and explanation
(0, 1/10, 9, 1/11, Yes, forall_n_ge_N)
Choose the least nonnegative N with |coefficient[point]|/(N+shift)<epsilon. The denominator increases thereafter, so every n≥N satisfies the strict bound. Equality at N is insufficient; a zero coefficient permits N=0. The exact result, retaining the requested order, is (0, 1/10, 9, 1/11, Yes, forall_n_ge_N).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-epsilon-witness · candidate
27. A Strict Pointwise Tail Witness Can Depend on the Point
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[1, 2], coefficients=[2, -3], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/11, point=1. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
View answer and explanation
(1, 1/11, 32, 3/34, Yes, forall_n_ge_N)
Choose the least nonnegative N with |coefficient[point]|/(N+shift)<epsilon. The denominator increases thereafter, so every n≥N satisfies the strict bound. Equality at N is insufficient; a zero coefficient permits N=0. The exact result, retaining the requested order, is (1, 1/11, 32, 3/34, Yes, forall_n_ge_N).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-epsilon-witness · candidate
28. A Strict Pointwise Tail Witness Can Depend on the Point
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[2, 3], coefficients=[3, -4], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/12, point=0. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
View answer and explanation
(0, 1/12, 35, 3/37, Yes, forall_n_ge_N)
Choose the least nonnegative N with |coefficient[point]|/(N+shift)<epsilon. The denominator increases thereafter, so every n≥N satisfies the strict bound. Equality at N is insufficient; a zero coefficient permits N=0. The exact result, retaining the requested order, is (0, 1/12, 35, 3/37, Yes, forall_n_ge_N).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-epsilon-witness · candidate
29. A Strict Pointwise Tail Witness Can Depend on the Point
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[3, 4], coefficients=[4, -5], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/13, point=1. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
View answer and explanation
(1, 1/13, 64, 5/66, Yes, forall_n_ge_N)
Choose the least nonnegative N with |coefficient[point]|/(N+shift)<epsilon. The denominator increases thereafter, so every n≥N satisfies the strict bound. Equality at N is insufficient; a zero coefficient permits N=0. The exact result, retaining the requested order, is (1, 1/13, 64, 5/66, Yes, forall_n_ge_N).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-epsilon-witness · candidate
30. A Strict Pointwise Tail Witness Can Depend on the Point
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[4, 5], coefficients=[5, -6], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/14, point=0. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
View answer and explanation
(0, 1/14, 69, 5/71, Yes, forall_n_ge_N)
Choose the least nonnegative N with |coefficient[point]|/(N+shift)<epsilon. The denominator increases thereafter, so every n≥N satisfies the strict bound. Equality at N is insufficient; a zero coefficient permits N=0. The exact result, retaining the requested order, is (0, 1/14, 69, 5/71, Yes, forall_n_ge_N).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-epsilon-witness · candidate
31. A Strict Pointwise Tail Witness Can Depend on the Point
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[5, 6], coefficients=[6, -7], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/15, point=1. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
View answer and explanation
(1, 1/15, 104, 7/106, Yes, forall_n_ge_N)
Choose the least nonnegative N with |coefficient[point]|/(N+shift)<epsilon. The denominator increases thereafter, so every n≥N satisfies the strict bound. Equality at N is insufficient; a zero coefficient permits N=0. The exact result, retaining the requested order, is (1, 1/15, 104, 7/106, Yes, forall_n_ge_N).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-epsilon-witness · candidate
32. A Strict Pointwise Tail Witness Can Depend on the Point
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[6, 7], coefficients=[7, -8], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/16, point=0. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
View answer and explanation
(0, 1/16, 111, 7/113, Yes, forall_n_ge_N)
Choose the least nonnegative N with |coefficient[point]|/(N+shift)<epsilon. The denominator increases thereafter, so every n≥N satisfies the strict bound. Equality at N is insufficient; a zero coefficient permits N=0. The exact result, retaining the requested order, is (0, 1/16, 111, 7/113, Yes, forall_n_ge_N).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-epsilon-witness · candidate
33. A Strict Pointwise Tail Witness Can Depend on the Point
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[7, 8], coefficients=[8, -9], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/17, point=1. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
View answer and explanation
(1, 1/17, 152, 9/154, Yes, forall_n_ge_N)
Choose the least nonnegative N with |coefficient[point]|/(N+shift)<epsilon. The denominator increases thereafter, so every n≥N satisfies the strict bound. Equality at N is insufficient; a zero coefficient permits N=0. The exact result, retaining the requested order, is (1, 1/17, 152, 9/154, Yes, forall_n_ge_N).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-epsilon-witness · candidate
34. A Strict Pointwise Tail Witness Can Depend on the Point
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[8, 9], coefficients=[9, -10], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/18, point=0. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
View answer and explanation
(0, 1/18, 161, 9/163, Yes, forall_n_ge_N)
Choose the least nonnegative N with |coefficient[point]|/(N+shift)<epsilon. The denominator increases thereafter, so every n≥N satisfies the strict bound. Equality at N is insufficient; a zero coefficient permits N=0. The exact result, retaining the requested order, is (0, 1/18, 161, 9/163, Yes, forall_n_ge_N).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-epsilon-witness · candidate
35. A Strict Pointwise Tail Witness Can Depend on the Point
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[9, 10], coefficients=[10, -11], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/19, point=1. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
View answer and explanation
(1, 1/19, 208, 11/210, Yes, forall_n_ge_N)
Choose the least nonnegative N with |coefficient[point]|/(N+shift)<epsilon. The denominator increases thereafter, so every n≥N satisfies the strict bound. Equality at N is insufficient; a zero coefficient permits N=0. The exact result, retaining the requested order, is (1, 1/19, 208, 11/210, Yes, forall_n_ge_N).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-epsilon-witness · candidate
36. A Strict Pointwise Tail Witness Can Depend on the Point
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[10, 11], coefficients=[11, -12], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/20, point=0. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
View answer and explanation
(0, 1/20, 219, 11/221, Yes, forall_n_ge_N)
Choose the least nonnegative N with |coefficient[point]|/(N+shift)<epsilon. The denominator increases thereafter, so every n≥N satisfies the strict bound. Equality at N is insufficient; a zero coefficient permits N=0. The exact result, retaining the requested order, is (0, 1/20, 219, 11/221, Yes, forall_n_ge_N).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-epsilon-witness · candidate
37. A Strict Pointwise Tail Witness Can Depend on the Point
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[11, 12], coefficients=[12, -13], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/21, point=1. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
View answer and explanation
(1, 1/21, 272, 13/274, Yes, forall_n_ge_N)
Choose the least nonnegative N with |coefficient[point]|/(N+shift)<epsilon. The denominator increases thereafter, so every n≥N satisfies the strict bound. Equality at N is insufficient; a zero coefficient permits N=0. The exact result, retaining the requested order, is (1, 1/21, 272, 13/274, Yes, forall_n_ge_N).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-epsilon-witness · candidate
38. A Strict Pointwise Tail Witness Can Depend on the Point
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[12, 13], coefficients=[13, -14], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/22, point=0. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
View answer and explanation
(0, 1/22, 285, 13/287, Yes, forall_n_ge_N)
Choose the least nonnegative N with |coefficient[point]|/(N+shift)<epsilon. The denominator increases thereafter, so every n≥N satisfies the strict bound. Equality at N is insufficient; a zero coefficient permits N=0. The exact result, retaining the requested order, is (0, 1/22, 285, 13/287, Yes, forall_n_ge_N).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-epsilon-witness · candidate
39. A Strict Pointwise Tail Witness Can Depend on the Point
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[13, 14], coefficients=[14, -15], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/23, point=1. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
View answer and explanation
(1, 1/23, 344, 15/346, Yes, forall_n_ge_N)
Choose the least nonnegative N with |coefficient[point]|/(N+shift)<epsilon. The denominator increases thereafter, so every n≥N satisfies the strict bound. Equality at N is insufficient; a zero coefficient permits N=0. The exact result, retaining the requested order, is (1, 1/23, 344, 15/346, Yes, forall_n_ge_N).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-epsilon-witness · candidate
40. A Strict Pointwise Tail Witness Can Depend on the Point
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[14, 15], coefficients=[15, -16], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/24, point=0. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
View answer and explanation
(0, 1/24, 359, 15/361, Yes, forall_n_ge_N)
Choose the least nonnegative N with |coefficient[point]|/(N+shift)<epsilon. The denominator increases thereafter, so every n≥N satisfies the strict bound. Equality at N is insufficient; a zero coefficient permits N=0. The exact result, retaining the requested order, is (0, 1/24, 359, 15/361, Yes, forall_n_ge_N).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-epsilon-witness · candidate
41. A Strict Pointwise Tail Witness Can Depend on the Point
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[15, 16], coefficients=[16, -17], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/25, point=1. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
View answer and explanation
(1, 1/25, 424, 17/426, Yes, forall_n_ge_N)
Choose the least nonnegative N with |coefficient[point]|/(N+shift)<epsilon. The denominator increases thereafter, so every n≥N satisfies the strict bound. Equality at N is insufficient; a zero coefficient permits N=0. The exact result, retaining the requested order, is (1, 1/25, 424, 17/426, Yes, forall_n_ge_N).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-epsilon-witness · candidate
42. A Strict Pointwise Tail Witness Can Depend on the Point
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[16, 17], coefficients=[17, -18], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/26, point=0. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
View answer and explanation
(0, 1/26, 441, 17/443, Yes, forall_n_ge_N)
Choose the least nonnegative N with |coefficient[point]|/(N+shift)<epsilon. The denominator increases thereafter, so every n≥N satisfies the strict bound. Equality at N is insufficient; a zero coefficient permits N=0. The exact result, retaining the requested order, is (0, 1/26, 441, 17/443, Yes, forall_n_ge_N).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-epsilon-witness · candidate
43. A Strict Pointwise Tail Witness Can Depend on the Point
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[17, 18], coefficients=[18, -19], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/27, point=1. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
View answer and explanation
(1, 1/27, 512, 19/514, Yes, forall_n_ge_N)
Choose the least nonnegative N with |coefficient[point]|/(N+shift)<epsilon. The denominator increases thereafter, so every n≥N satisfies the strict bound. Equality at N is insufficient; a zero coefficient permits N=0. The exact result, retaining the requested order, is (1, 1/27, 512, 19/514, Yes, forall_n_ge_N).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-epsilon-witness · candidate
44. A Strict Pointwise Tail Witness Can Depend on the Point
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[18, 19], coefficients=[19, -20], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/28, point=0. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
View answer and explanation
(0, 1/28, 531, 19/533, Yes, forall_n_ge_N)
Choose the least nonnegative N with |coefficient[point]|/(N+shift)<epsilon. The denominator increases thereafter, so every n≥N satisfies the strict bound. Equality at N is insufficient; a zero coefficient permits N=0. The exact result, retaining the requested order, is (0, 1/28, 531, 19/533, Yes, forall_n_ge_N).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-epsilon-witness · candidate
45. A Strict Pointwise Tail Witness Can Depend on the Point
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[19, 20], coefficients=[20, -21], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/29, point=1. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
View answer and explanation
(1, 1/29, 608, 21/610, Yes, forall_n_ge_N)
Choose the least nonnegative N with |coefficient[point]|/(N+shift)<epsilon. The denominator increases thereafter, so every n≥N satisfies the strict bound. Equality at N is insufficient; a zero coefficient permits N=0. The exact result, retaining the requested order, is (1, 1/29, 608, 21/610, Yes, forall_n_ge_N).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-epsilon-witness · candidate
46. A Strict Pointwise Tail Witness Can Depend on the Point
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[20, 21], coefficients=[21, -22], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/30, point=0. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
View answer and explanation
(0, 1/30, 629, 21/631, Yes, forall_n_ge_N)
Choose the least nonnegative N with |coefficient[point]|/(N+shift)<epsilon. The denominator increases thereafter, so every n≥N satisfies the strict bound. Equality at N is insufficient; a zero coefficient permits N=0. The exact result, retaining the requested order, is (0, 1/30, 629, 21/631, Yes, forall_n_ge_N).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-epsilon-witness · candidate
47. A Strict Pointwise Tail Witness Can Depend on the Point
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[21, 22], coefficients=[22, -23], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/31, point=1. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
View answer and explanation
(1, 1/31, 712, 23/714, Yes, forall_n_ge_N)
Choose the least nonnegative N with |coefficient[point]|/(N+shift)<epsilon. The denominator increases thereafter, so every n≥N satisfies the strict bound. Equality at N is insufficient; a zero coefficient permits N=0. The exact result, retaining the requested order, is (1, 1/31, 712, 23/714, Yes, forall_n_ge_N).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-epsilon-witness · candidate
48. A Strict Pointwise Tail Witness Can Depend on the Point
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[22, 23], coefficients=[23, -24], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/32, point=0. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
View answer and explanation
(0, 1/32, 735, 23/737, Yes, forall_n_ge_N)
Choose the least nonnegative N with |coefficient[point]|/(N+shift)<epsilon. The denominator increases thereafter, so every n≥N satisfies the strict bound. Equality at N is insufficient; a zero coefficient permits N=0. The exact result, retaining the requested order, is (0, 1/32, 735, 23/737, Yes, forall_n_ge_N).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-epsilon-witness · candidate
49. A Strict Pointwise Tail Witness Can Depend on the Point
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[23, 24], coefficients=[24, -25], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/33, point=1. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
View answer and explanation
(1, 1/33, 824, 25/826, Yes, forall_n_ge_N)
Choose the least nonnegative N with |coefficient[point]|/(N+shift)<epsilon. The denominator increases thereafter, so every n≥N satisfies the strict bound. Equality at N is insufficient; a zero coefficient permits N=0. The exact result, retaining the requested order, is (1, 1/33, 824, 25/826, Yes, forall_n_ge_N).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-epsilon-witness · candidate
50. A Strict Pointwise Tail Witness Can Depend on the Point
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[24, 25], coefficients=[25, -26], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/34, point=0. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
View answer and explanation
(0, 1/34, 849, 25/851, Yes, forall_n_ge_N)
Choose the least nonnegative N with |coefficient[point]|/(N+shift)<epsilon. The denominator increases thereafter, so every n≥N satisfies the strict bound. Equality at N is insufficient; a zero coefficient permits N=0. The exact result, retaining the requested order, is (0, 1/34, 849, 25/851, Yes, forall_n_ge_N).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-epsilon-witness · candidate
51. Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[0, 1], coefficients=[1, -2], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/10. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.
View answer and explanation
(1/10, [[0, 9], [1, 19]], 19, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)
Compute each strict pointwise witness and take the maximum over this complete finite domain. This one N works for every listed point. The argument does not justify interchanging the quantifiers on an arbitrary infinite domain. The exact result, retaining the requested order, is (1/10, [[0, 9], [1, 19]], 19, Yes, finite_domain_maximum_only_not_infinite_domain_theorem).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-uniform-max · candidate
52. Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[1, 2], coefficients=[2, -3], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/11. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.
View answer and explanation
(1/11, [[0, 21], [1, 32]], 32, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)
Compute each strict pointwise witness and take the maximum over this complete finite domain. This one N works for every listed point. The argument does not justify interchanging the quantifiers on an arbitrary infinite domain. The exact result, retaining the requested order, is (1/11, [[0, 21], [1, 32]], 32, Yes, finite_domain_maximum_only_not_infinite_domain_theorem).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-uniform-max · candidate
53. Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[2, 3], coefficients=[3, -4], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/12. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.
View answer and explanation
(1/12, [[0, 35], [1, 47]], 47, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)
Compute each strict pointwise witness and take the maximum over this complete finite domain. This one N works for every listed point. The argument does not justify interchanging the quantifiers on an arbitrary infinite domain. The exact result, retaining the requested order, is (1/12, [[0, 35], [1, 47]], 47, Yes, finite_domain_maximum_only_not_infinite_domain_theorem).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-uniform-max · candidate
54. Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[3, 4], coefficients=[4, -5], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/13. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.
View answer and explanation
(1/13, [[0, 51], [1, 64]], 64, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)
Compute each strict pointwise witness and take the maximum over this complete finite domain. This one N works for every listed point. The argument does not justify interchanging the quantifiers on an arbitrary infinite domain. The exact result, retaining the requested order, is (1/13, [[0, 51], [1, 64]], 64, Yes, finite_domain_maximum_only_not_infinite_domain_theorem).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-uniform-max · candidate
55. Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[4, 5], coefficients=[5, -6], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/14. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.
View answer and explanation
(1/14, [[0, 69], [1, 83]], 83, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)
Compute each strict pointwise witness and take the maximum over this complete finite domain. This one N works for every listed point. The argument does not justify interchanging the quantifiers on an arbitrary infinite domain. The exact result, retaining the requested order, is (1/14, [[0, 69], [1, 83]], 83, Yes, finite_domain_maximum_only_not_infinite_domain_theorem).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-uniform-max · candidate
56. Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[5, 6], coefficients=[6, -7], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/15. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.
View answer and explanation
(1/15, [[0, 89], [1, 104]], 104, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)
Compute each strict pointwise witness and take the maximum over this complete finite domain. This one N works for every listed point. The argument does not justify interchanging the quantifiers on an arbitrary infinite domain. The exact result, retaining the requested order, is (1/15, [[0, 89], [1, 104]], 104, Yes, finite_domain_maximum_only_not_infinite_domain_theorem).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-uniform-max · candidate
57. Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[6, 7], coefficients=[7, -8], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/16. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.
View answer and explanation
(1/16, [[0, 111], [1, 127]], 127, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)
Compute each strict pointwise witness and take the maximum over this complete finite domain. This one N works for every listed point. The argument does not justify interchanging the quantifiers on an arbitrary infinite domain. The exact result, retaining the requested order, is (1/16, [[0, 111], [1, 127]], 127, Yes, finite_domain_maximum_only_not_infinite_domain_theorem).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-uniform-max · candidate
58. Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[7, 8], coefficients=[8, -9], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/17. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.
View answer and explanation
(1/17, [[0, 135], [1, 152]], 152, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)
Compute each strict pointwise witness and take the maximum over this complete finite domain. This one N works for every listed point. The argument does not justify interchanging the quantifiers on an arbitrary infinite domain. The exact result, retaining the requested order, is (1/17, [[0, 135], [1, 152]], 152, Yes, finite_domain_maximum_only_not_infinite_domain_theorem).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-uniform-max · candidate
59. Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[8, 9], coefficients=[9, -10], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/18. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.
View answer and explanation
(1/18, [[0, 161], [1, 179]], 179, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)
Compute each strict pointwise witness and take the maximum over this complete finite domain. This one N works for every listed point. The argument does not justify interchanging the quantifiers on an arbitrary infinite domain. The exact result, retaining the requested order, is (1/18, [[0, 161], [1, 179]], 179, Yes, finite_domain_maximum_only_not_infinite_domain_theorem).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-uniform-max · candidate
60. Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[9, 10], coefficients=[10, -11], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/19. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.
View answer and explanation
(1/19, [[0, 189], [1, 208]], 208, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)
Compute each strict pointwise witness and take the maximum over this complete finite domain. This one N works for every listed point. The argument does not justify interchanging the quantifiers on an arbitrary infinite domain. The exact result, retaining the requested order, is (1/19, [[0, 189], [1, 208]], 208, Yes, finite_domain_maximum_only_not_infinite_domain_theorem).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-uniform-max · candidate
61. Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[10, 11], coefficients=[11, -12], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/20. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.
View answer and explanation
(1/20, [[0, 219], [1, 239]], 239, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)
Compute each strict pointwise witness and take the maximum over this complete finite domain. This one N works for every listed point. The argument does not justify interchanging the quantifiers on an arbitrary infinite domain. The exact result, retaining the requested order, is (1/20, [[0, 219], [1, 239]], 239, Yes, finite_domain_maximum_only_not_infinite_domain_theorem).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-uniform-max · candidate
62. Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[11, 12], coefficients=[12, -13], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/21. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.
View answer and explanation
(1/21, [[0, 251], [1, 272]], 272, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)
Compute each strict pointwise witness and take the maximum over this complete finite domain. This one N works for every listed point. The argument does not justify interchanging the quantifiers on an arbitrary infinite domain. The exact result, retaining the requested order, is (1/21, [[0, 251], [1, 272]], 272, Yes, finite_domain_maximum_only_not_infinite_domain_theorem).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-uniform-max · candidate
63. Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[12, 13], coefficients=[13, -14], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/22. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.
View answer and explanation
(1/22, [[0, 285], [1, 307]], 307, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)
Compute each strict pointwise witness and take the maximum over this complete finite domain. This one N works for every listed point. The argument does not justify interchanging the quantifiers on an arbitrary infinite domain. The exact result, retaining the requested order, is (1/22, [[0, 285], [1, 307]], 307, Yes, finite_domain_maximum_only_not_infinite_domain_theorem).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-uniform-max · candidate
64. Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[13, 14], coefficients=[14, -15], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/23. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.
View answer and explanation
(1/23, [[0, 321], [1, 344]], 344, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)
Compute each strict pointwise witness and take the maximum over this complete finite domain. This one N works for every listed point. The argument does not justify interchanging the quantifiers on an arbitrary infinite domain. The exact result, retaining the requested order, is (1/23, [[0, 321], [1, 344]], 344, Yes, finite_domain_maximum_only_not_infinite_domain_theorem).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-uniform-max · candidate
65. Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[14, 15], coefficients=[15, -16], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/24. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.
View answer and explanation
(1/24, [[0, 359], [1, 383]], 383, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)
Compute each strict pointwise witness and take the maximum over this complete finite domain. This one N works for every listed point. The argument does not justify interchanging the quantifiers on an arbitrary infinite domain. The exact result, retaining the requested order, is (1/24, [[0, 359], [1, 383]], 383, Yes, finite_domain_maximum_only_not_infinite_domain_theorem).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-uniform-max · candidate
66. Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[15, 16], coefficients=[16, -17], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/25. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.
View answer and explanation
(1/25, [[0, 399], [1, 424]], 424, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)
Compute each strict pointwise witness and take the maximum over this complete finite domain. This one N works for every listed point. The argument does not justify interchanging the quantifiers on an arbitrary infinite domain. The exact result, retaining the requested order, is (1/25, [[0, 399], [1, 424]], 424, Yes, finite_domain_maximum_only_not_infinite_domain_theorem).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-uniform-max · candidate
67. Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[16, 17], coefficients=[17, -18], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/26. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.
View answer and explanation
(1/26, [[0, 441], [1, 467]], 467, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)
Compute each strict pointwise witness and take the maximum over this complete finite domain. This one N works for every listed point. The argument does not justify interchanging the quantifiers on an arbitrary infinite domain. The exact result, retaining the requested order, is (1/26, [[0, 441], [1, 467]], 467, Yes, finite_domain_maximum_only_not_infinite_domain_theorem).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-uniform-max · candidate
68. Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[17, 18], coefficients=[18, -19], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/27. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.
View answer and explanation
(1/27, [[0, 485], [1, 512]], 512, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)
Compute each strict pointwise witness and take the maximum over this complete finite domain. This one N works for every listed point. The argument does not justify interchanging the quantifiers on an arbitrary infinite domain. The exact result, retaining the requested order, is (1/27, [[0, 485], [1, 512]], 512, Yes, finite_domain_maximum_only_not_infinite_domain_theorem).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-uniform-max · candidate
69. Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[18, 19], coefficients=[19, -20], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/28. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.
View answer and explanation
(1/28, [[0, 531], [1, 559]], 559, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)
Compute each strict pointwise witness and take the maximum over this complete finite domain. This one N works for every listed point. The argument does not justify interchanging the quantifiers on an arbitrary infinite domain. The exact result, retaining the requested order, is (1/28, [[0, 531], [1, 559]], 559, Yes, finite_domain_maximum_only_not_infinite_domain_theorem).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-uniform-max · candidate
70. Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[19, 20], coefficients=[20, -21], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/29. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.
View answer and explanation
(1/29, [[0, 579], [1, 608]], 608, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)
Compute each strict pointwise witness and take the maximum over this complete finite domain. This one N works for every listed point. The argument does not justify interchanging the quantifiers on an arbitrary infinite domain. The exact result, retaining the requested order, is (1/29, [[0, 579], [1, 608]], 608, Yes, finite_domain_maximum_only_not_infinite_domain_theorem).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-uniform-max · candidate
71. Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[20, 21], coefficients=[21, -22], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/30. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.
View answer and explanation
(1/30, [[0, 629], [1, 659]], 659, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)
Compute each strict pointwise witness and take the maximum over this complete finite domain. This one N works for every listed point. The argument does not justify interchanging the quantifiers on an arbitrary infinite domain. The exact result, retaining the requested order, is (1/30, [[0, 629], [1, 659]], 659, Yes, finite_domain_maximum_only_not_infinite_domain_theorem).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-uniform-max · candidate
72. Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[21, 22], coefficients=[22, -23], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/31. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.
View answer and explanation
(1/31, [[0, 681], [1, 712]], 712, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)
Compute each strict pointwise witness and take the maximum over this complete finite domain. This one N works for every listed point. The argument does not justify interchanging the quantifiers on an arbitrary infinite domain. The exact result, retaining the requested order, is (1/31, [[0, 681], [1, 712]], 712, Yes, finite_domain_maximum_only_not_infinite_domain_theorem).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-uniform-max · candidate
73. Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[22, 23], coefficients=[23, -24], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/32. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.
View answer and explanation
(1/32, [[0, 735], [1, 767]], 767, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)
Compute each strict pointwise witness and take the maximum over this complete finite domain. This one N works for every listed point. The argument does not justify interchanging the quantifiers on an arbitrary infinite domain. The exact result, retaining the requested order, is (1/32, [[0, 735], [1, 767]], 767, Yes, finite_domain_maximum_only_not_infinite_domain_theorem).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-uniform-max · candidate
74. Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[23, 24], coefficients=[24, -25], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/33. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.
View answer and explanation
(1/33, [[0, 791], [1, 824]], 824, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)
Compute each strict pointwise witness and take the maximum over this complete finite domain. This one N works for every listed point. The argument does not justify interchanging the quantifiers on an arbitrary infinite domain. The exact result, retaining the requested order, is (1/33, [[0, 791], [1, 824]], 824, Yes, finite_domain_maximum_only_not_infinite_domain_theorem).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-uniform-max · candidate
75. Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[24, 25], coefficients=[25, -26], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/34. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.
View answer and explanation
(1/34, [[0, 849], [1, 883]], 883, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)
Compute each strict pointwise witness and take the maximum over this complete finite domain. This one N works for every listed point. The argument does not justify interchanging the quantifiers on an arbitrary infinite domain. The exact result, retaining the requested order, is (1/34, [[0, 849], [1, 883]], 883, Yes, finite_domain_maximum_only_not_infinite_domain_theorem).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-domain-uniform-max · candidate
76. A Long Prefix Still Does Not Replace the All-n Tail
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[0, 1], coefficients=[1, -2], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=1, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].
View answer and explanation
(1, [[0, [1/2]], [1, [0]]], No, finite_prefix_not_pointwise_convergence_certificate)
Display only the requested finite values. The No sufficiency flag concerns inference from this prefix alone, not a claim that the separately declared symbolic tail fails to converge. The exact result, retaining the requested order, is (1, [[0, [1/2]], [1, [0]]], No, finite_prefix_not_pointwise_convergence_certificate).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-prefix-boundary · candidate
77. A Long Prefix Still Does Not Replace the All-n Tail
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[1, 2], coefficients=[2, -3], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=2, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].
View answer and explanation
(2, [[0, [2, 5/3]], [1, [1/2, 1]]], No, finite_prefix_not_pointwise_convergence_certificate)
Display only the requested finite values. The No sufficiency flag concerns inference from this prefix alone, not a claim that the separately declared symbolic tail fails to converge. The exact result, retaining the requested order, is (2, [[0, [2, 5/3]], [1, [1/2, 1]]], No, finite_prefix_not_pointwise_convergence_certificate).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-prefix-boundary · candidate
78. A Long Prefix Still Does Not Replace the All-n Tail
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[2, 3], coefficients=[3, -4], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=3, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].
View answer and explanation
(3, [[0, [7/2, 3, 11/4]], [1, [1, 5/3, 2]]], No, finite_prefix_not_pointwise_convergence_certificate)
Display only the requested finite values. The No sufficiency flag concerns inference from this prefix alone, not a claim that the separately declared symbolic tail fails to converge. The exact result, retaining the requested order, is (3, [[0, [7/2, 3, 11/4]], [1, [1, 5/3, 2]]], No, finite_prefix_not_pointwise_convergence_certificate).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-prefix-boundary · candidate
79. A Long Prefix Still Does Not Replace the All-n Tail
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[3, 4], coefficients=[4, -5], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=4, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].
View answer and explanation
(4, [[0, [5, 13/3, 4, 19/5]], [1, [3/2, 7/3, 11/4, 3]]], No, finite_prefix_not_pointwise_convergence_certificate)
Display only the requested finite values. The No sufficiency flag concerns inference from this prefix alone, not a claim that the separately declared symbolic tail fails to converge. The exact result, retaining the requested order, is (4, [[0, [5, 13/3, 4, 19/5]], [1, [3/2, 7/3, 11/4, 3]]], No, finite_prefix_not_pointwise_convergence_certificate).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-prefix-boundary · candidate
80. A Long Prefix Still Does Not Replace the All-n Tail
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[4, 5], coefficients=[5, -6], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=5, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].
View answer and explanation
(5, [[0, [13/2, 17/3, 21/4, 5, 29/6]], [1, [2, 3, 7/2, 19/5, 4]]], No, finite_prefix_not_pointwise_convergence_certificate)
Display only the requested finite values. The No sufficiency flag concerns inference from this prefix alone, not a claim that the separately declared symbolic tail fails to converge. The exact result, retaining the requested order, is (5, [[0, [13/2, 17/3, 21/4, 5, 29/6]], [1, [2, 3, 7/2, 19/5, 4]]], No, finite_prefix_not_pointwise_convergence_certificate).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-prefix-boundary · candidate
81. A Long Prefix Still Does Not Replace the All-n Tail
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[5, 6], coefficients=[6, -7], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=6, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].
View answer and explanation
(6, [[0, [8, 7, 13/2, 31/5, 6, 41/7]], [1, [5/2, 11/3, 17/4, 23/5, 29/6, 5]]], No, finite_prefix_not_pointwise_convergence_certificate)
Display only the requested finite values. The No sufficiency flag concerns inference from this prefix alone, not a claim that the separately declared symbolic tail fails to converge. The exact result, retaining the requested order, is (6, [[0, [8, 7, 13/2, 31/5, 6, 41/7]], [1, [5/2, 11/3, 17/4, 23/5, 29/6, 5]]], No, finite_prefix_not_pointwise_convergence_certificate).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-prefix-boundary · candidate
82. A Long Prefix Still Does Not Replace the All-n Tail
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[6, 7], coefficients=[7, -8], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=7, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].
View answer and explanation
(7, [[0, [19/2, 25/3, 31/4, 37/5, 43/6, 7, 55/8]], [1, [3, 13/3, 5, 27/5, 17/3, 41/7, 6]]], No, finite_prefix_not_pointwise_convergence_certificate)
Display only the requested finite values. The No sufficiency flag concerns inference from this prefix alone, not a claim that the separately declared symbolic tail fails to converge. The exact result, retaining the requested order, is (7, [[0, [19/2, 25/3, 31/4, 37/5, 43/6, 7, 55/8]], [1, [3, 13/3, 5, 27/5, 17/3, 41/7, 6]]], No, finite_prefix_not_pointwise_convergence_certificate).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-prefix-boundary · candidate
83. A Long Prefix Still Does Not Replace the All-n Tail
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[7, 8], coefficients=[8, -9], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=8, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].
View answer and explanation
(8, [[0, [11, 29/3, 9, 43/5, 25/3, 57/7, 8, 71/9]], [1, [7/2, 5, 23/4, 31/5, 13/2, 47/7, 55/8, 7]]], No, finite_prefix_not_pointwise_convergence_certificate)
Display only the requested finite values. The No sufficiency flag concerns inference from this prefix alone, not a claim that the separately declared symbolic tail fails to converge. The exact result, retaining the requested order, is (8, [[0, [11, 29/3, 9, 43/5, 25/3, 57/7, 8, 71/9]], [1, [7/2, 5, 23/4, 31/5, 13/2, 47/7, 55/8, 7]]], No, finite_prefix_not_pointwise_convergence_certificate).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-prefix-boundary · candidate
84. A Long Prefix Still Does Not Replace the All-n Tail
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[8, 9], coefficients=[9, -10], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=9, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].
View answer and explanation
(9, [[0, [25/2, 11, 41/4, 49/5, 19/2, 65/7, 73/8, 9, 89/10]], [1, [4, 17/3, 13/2, 7, 22/3, 53/7, 31/4, 71/9, 8]]], No, finite_prefix_not_pointwise_convergence_certificate)
Display only the requested finite values. The No sufficiency flag concerns inference from this prefix alone, not a claim that the separately declared symbolic tail fails to converge. The exact result, retaining the requested order, is (9, [[0, [25/2, 11, 41/4, 49/5, 19/2, 65/7, 73/8, 9, 89/10]], [1, [4, 17/3, 13/2, 7, 22/3, 53/7, 31/4, 71/9, 8]]], No, finite_prefix_not_pointwise_convergence_certificate).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-prefix-boundary · candidate
85. A Long Prefix Still Does Not Replace the All-n Tail
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[9, 10], coefficients=[10, -11], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=10, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].
View answer and explanation
(10, [[0, [14, 37/3, 23/2, 11, 32/3, 73/7, 41/4, 91/9, 10, 109/11]], [1, [9/2, 19/3, 29/4, 39/5, 49/6, 59/7, 69/8, 79/9, 89/10, 9]]], No, finite_prefix_not_pointwise_convergence_certificate)
Display only the requested finite values. The No sufficiency flag concerns inference from this prefix alone, not a claim that the separately declared symbolic tail fails to converge. The exact result, retaining the requested order, is (10, [[0, [14, 37/3, 23/2, 11, 32/3, 73/7, 41/4, 91/9, 10, 109/11]], [1, [9/2, 19/3, 29/4, 39/5, 49/6, 59/7, 69/8, 79/9, 89/10, 9]]], No, finite_prefix_not_pointwise_convergence_certificate).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-prefix-boundary · candidate
86. A Long Prefix Still Does Not Replace the All-n Tail
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[10, 11], coefficients=[11, -12], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=11, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].
View answer and explanation
(11, [[0, [31/2, 41/3, 51/4, 61/5, 71/6, 81/7, 91/8, 101/9, 111/10, 11, 131/12]], [1, [5, 7, 8, 43/5, 9, 65/7, 19/2, 29/3, 49/5, 109/11, 10]]], No, finite_prefix_not_pointwise_convergence_certificate)
Display only the requested finite values. The No sufficiency flag concerns inference from this prefix alone, not a claim that the separately declared symbolic tail fails to converge. The exact result, retaining the requested order, is (11, [[0, [31/2, 41/3, 51/4, 61/5, 71/6, 81/7, 91/8, 101/9, 111/10, 11, 131/12]], [1, [5, 7, 8, 43/5, 9, 65/7, 19/2, 29/3, 49/5, 109/11, 10]]], No, finite_prefix_not_pointwise_convergence_certificate).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-prefix-boundary · candidate
87. A Long Prefix Still Does Not Replace the All-n Tail
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[11, 12], coefficients=[12, -13], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=12, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].
View answer and explanation
(12, [[0, [17, 15, 14, 67/5, 13, 89/7, 25/2, 37/3, 61/5, 133/11, 12, 155/13]], [1, [11/2, 23/3, 35/4, 47/5, 59/6, 71/7, 83/8, 95/9, 107/10, 119/11, 131/12, 11]]], No, finite_prefix_not_pointwise_convergence_certificate)
Display only the requested finite values. The No sufficiency flag concerns inference from this prefix alone, not a claim that the separately declared symbolic tail fails to converge. The exact result, retaining the requested order, is (12, [[0, [17, 15, 14, 67/5, 13, 89/7, 25/2, 37/3, 61/5, 133/11, 12, 155/13]], [1, [11/2, 23/3, 35/4, 47/5, 59/6, 71/7, 83/8, 95/9, 107/10, 119/11, 131/12, 11]]], No, finite_prefix_not_pointwise_convergence_certificate).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-prefix-boundary · candidate
88. A Long Prefix Still Does Not Replace the All-n Tail
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[12, 13], coefficients=[13, -14], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=13, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].
View answer and explanation
(13, [[0, [37/2, 49/3, 61/4, 73/5, 85/6, 97/7, 109/8, 121/9, 133/10, 145/11, 157/12, 13, 181/14]], [1, [6, 25/3, 19/2, 51/5, 32/3, 11, 45/4, 103/9, 58/5, 129/11, 71/6, 155/13, 12]]], No, finite_prefix_not_pointwise_convergence_certificate)
Display only the requested finite values. The No sufficiency flag concerns inference from this prefix alone, not a claim that the separately declared symbolic tail fails to converge. The exact result, retaining the requested order, is (13, [[0, [37/2, 49/3, 61/4, 73/5, 85/6, 97/7, 109/8, 121/9, 133/10, 145/11, 157/12, 13, 181/14]], [1, [6, 25/3, 19/2, 51/5, 32/3, 11, 45/4, 103/9, 58/5, 129/11, 71/6, 155/13, 12]]], No, finite_prefix_not_pointwise_convergence_certificate).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-prefix-boundary · candidate
89. A Long Prefix Still Does Not Replace the All-n Tail
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[13, 14], coefficients=[14, -15], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=14, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].
View answer and explanation
(14, [[0, [20, 53/3, 33/2, 79/5, 46/3, 15, 59/4, 131/9, 72/5, 157/11, 85/6, 183/13, 14, 209/15]], [1, [13/2, 9, 41/4, 11, 23/2, 83/7, 97/8, 37/3, 25/2, 139/11, 51/4, 167/13, 181/14, 13]]], No, finite_prefix_not_pointwise_convergence_certificate)
Display only the requested finite values. The No sufficiency flag concerns inference from this prefix alone, not a claim that the separately declared symbolic tail fails to converge. The exact result, retaining the requested order, is (14, [[0, [20, 53/3, 33/2, 79/5, 46/3, 15, 59/4, 131/9, 72/5, 157/11, 85/6, 183/13, 14, 209/15]], [1, [13/2, 9, 41/4, 11, 23/2, 83/7, 97/8, 37/3, 25/2, 139/11, 51/4, 167/13, 181/14, 13]]], No, finite_prefix_not_pointwise_convergence_certificate).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-prefix-boundary · candidate
90. A Long Prefix Still Does Not Replace the All-n Tail
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[14, 15], coefficients=[15, -16], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=15, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].
View answer and explanation
(15, [[0, [43/2, 19, 71/4, 17, 33/2, 113/7, 127/8, 47/3, 31/2, 169/11, 61/4, 197/13, 211/14, 15, 239/16]], [1, [7, 29/3, 11, 59/5, 37/3, 89/7, 13, 119/9, 67/5, 149/11, 41/3, 179/13, 97/7, 209/15, 14]]], No, finite_prefix_not_pointwise_convergence_certificate)
Display only the requested finite values. The No sufficiency flag concerns inference from this prefix alone, not a claim that the separately declared symbolic tail fails to converge. The exact result, retaining the requested order, is (15, [[0, [43/2, 19, 71/4, 17, 33/2, 113/7, 127/8, 47/3, 31/2, 169/11, 61/4, 197/13, 211/14, 15, 239/16]], [1, [7, 29/3, 11, 59/5, 37/3, 89/7, 13, 119/9, 67/5, 149/11, 41/3, 179/13, 97/7, 209/15, 14]]], No, finite_prefix_not_pointwise_convergence_certificate).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-prefix-boundary · candidate
91. A Long Prefix Still Does Not Replace the All-n Tail
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[15, 16], coefficients=[16, -17], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=16, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].
View answer and explanation
(16, [[0, [23, 61/3, 19, 91/5, 53/3, 121/7, 17, 151/9, 83/5, 181/11, 49/3, 211/13, 113/7, 241/15, 16, 271/17]], [1, [15/2, 31/3, 47/4, 63/5, 79/6, 95/7, 111/8, 127/9, 143/10, 159/11, 175/12, 191/13, 207/14, 223/15, 239/16, 15]]], No, finite_prefix_not_pointwise_convergence_certificate)
Display only the requested finite values. The No sufficiency flag concerns inference from this prefix alone, not a claim that the separately declared symbolic tail fails to converge. The exact result, retaining the requested order, is (16, [[0, [23, 61/3, 19, 91/5, 53/3, 121/7, 17, 151/9, 83/5, 181/11, 49/3, 211/13, 113/7, 241/15, 16, 271/17]], [1, [15/2, 31/3, 47/4, 63/5, 79/6, 95/7, 111/8, 127/9, 143/10, 159/11, 175/12, 191/13, 207/14, 223/15, 239/16, 15]]], No, finite_prefix_not_pointwise_convergence_certificate).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-prefix-boundary · candidate
92. A Long Prefix Still Does Not Replace the All-n Tail
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[16, 17], coefficients=[17, -18], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=17, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].
View answer and explanation
(17, [[0, [49/2, 65/3, 81/4, 97/5, 113/6, 129/7, 145/8, 161/9, 177/10, 193/11, 209/12, 225/13, 241/14, 257/15, 273/16, 17, 305/18]], [1, [8, 11, 25/2, 67/5, 14, 101/7, 59/4, 15, 76/5, 169/11, 31/2, 203/13, 110/7, 79/5, 127/8, 271/17, 16]]], No, finite_prefix_not_pointwise_convergence_certificate)
Display only the requested finite values. The No sufficiency flag concerns inference from this prefix alone, not a claim that the separately declared symbolic tail fails to converge. The exact result, retaining the requested order, is (17, [[0, [49/2, 65/3, 81/4, 97/5, 113/6, 129/7, 145/8, 161/9, 177/10, 193/11, 209/12, 225/13, 241/14, 257/15, 273/16, 17, 305/18]], [1, [8, 11, 25/2, 67/5, 14, 101/7, 59/4, 15, 76/5, 169/11, 31/2, 203/13, 110/7, 79/5, 127/8, 271/17, 16]]], No, finite_prefix_not_pointwise_convergence_certificate).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-prefix-boundary · candidate
93. A Long Prefix Still Does Not Replace the All-n Tail
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[17, 18], coefficients=[18, -19], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=18, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].
View answer and explanation
(18, [[0, [26, 23, 43/2, 103/5, 20, 137/7, 77/4, 19, 94/5, 205/11, 37/2, 239/13, 128/7, 91/5, 145/8, 307/17, 18, 341/19]], [1, [17/2, 35/3, 53/4, 71/5, 89/6, 107/7, 125/8, 143/9, 161/10, 179/11, 197/12, 215/13, 233/14, 251/15, 269/16, 287/17, 305/18, 17]]], No, finite_prefix_not_pointwise_convergence_certificate)
Display only the requested finite values. The No sufficiency flag concerns inference from this prefix alone, not a claim that the separately declared symbolic tail fails to converge. The exact result, retaining the requested order, is (18, [[0, [26, 23, 43/2, 103/5, 20, 137/7, 77/4, 19, 94/5, 205/11, 37/2, 239/13, 128/7, 91/5, 145/8, 307/17, 18, 341/19]], [1, [17/2, 35/3, 53/4, 71/5, 89/6, 107/7, 125/8, 143/9, 161/10, 179/11, 197/12, 215/13, 233/14, 251/15, 269/16, 287/17, 305/18, 17]]], No, finite_prefix_not_pointwise_convergence_certificate).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-prefix-boundary · candidate
94. A Long Prefix Still Does Not Replace the All-n Tail
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[18, 19], coefficients=[19, -20], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=19, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].
View answer and explanation
(19, [[0, [55/2, 73/3, 91/4, 109/5, 127/6, 145/7, 163/8, 181/9, 199/10, 217/11, 235/12, 253/13, 271/14, 289/15, 307/16, 325/17, 343/18, 19, 379/20]], [1, [9, 37/3, 14, 15, 47/3, 113/7, 33/2, 151/9, 17, 189/11, 52/3, 227/13, 123/7, 53/3, 71/4, 303/17, 161/9, 341/19, 18]]], No, finite_prefix_not_pointwise_convergence_certificate)
Display only the requested finite values. The No sufficiency flag concerns inference from this prefix alone, not a claim that the separately declared symbolic tail fails to converge. The exact result, retaining the requested order, is (19, [[0, [55/2, 73/3, 91/4, 109/5, 127/6, 145/7, 163/8, 181/9, 199/10, 217/11, 235/12, 253/13, 271/14, 289/15, 307/16, 325/17, 343/18, 19, 379/20]], [1, [9, 37/3, 14, 15, 47/3, 113/7, 33/2, 151/9, 17, 189/11, 52/3, 227/13, 123/7, 53/3, 71/4, 303/17, 161/9, 341/19, 18]]], No, finite_prefix_not_pointwise_convergence_certificate).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-prefix-boundary · candidate
95. A Long Prefix Still Does Not Replace the All-n Tail
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[19, 20], coefficients=[20, -21], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=20, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].
View answer and explanation
(20, [[0, [29, 77/3, 24, 23, 67/3, 153/7, 43/2, 191/9, 21, 229/11, 62/3, 267/13, 143/7, 61/3, 81/4, 343/17, 181/9, 381/19, 20, 419/21]], [1, [19/2, 13, 59/4, 79/5, 33/2, 17, 139/8, 53/3, 179/10, 199/11, 73/4, 239/13, 37/2, 93/5, 299/16, 319/17, 113/6, 359/19, 379/20, 19]]], No, finite_prefix_not_pointwise_convergence_certificate)
Display only the requested finite values. The No sufficiency flag concerns inference from this prefix alone, not a claim that the separately declared symbolic tail fails to converge. The exact result, retaining the requested order, is (20, [[0, [29, 77/3, 24, 23, 67/3, 153/7, 43/2, 191/9, 21, 229/11, 62/3, 267/13, 143/7, 61/3, 81/4, 343/17, 181/9, 381/19, 20, 419/21]], [1, [19/2, 13, 59/4, 79/5, 33/2, 17, 139/8, 53/3, 179/10, 199/11, 73/4, 239/13, 37/2, 93/5, 299/16, 319/17, 113/6, 359/19, 379/20, 19]]], No, finite_prefix_not_pointwise_convergence_certificate).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-prefix-boundary · candidate
96. A Long Prefix Still Does Not Replace the All-n Tail
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[20, 21], coefficients=[21, -22], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=21, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].
View answer and explanation
(21, [[0, [61/2, 27, 101/4, 121/5, 47/2, 23, 181/8, 67/3, 221/10, 241/11, 87/4, 281/13, 43/2, 107/5, 341/16, 361/17, 127/6, 401/19, 421/20, 21, 461/22]], [1, [10, 41/3, 31/2, 83/5, 52/3, 125/7, 73/4, 167/9, 94/5, 19, 115/6, 251/13, 136/7, 293/15, 157/8, 335/17, 178/9, 377/19, 199/10, 419/21, 20]]], No, finite_prefix_not_pointwise_convergence_certificate)
Display only the requested finite values. The No sufficiency flag concerns inference from this prefix alone, not a claim that the separately declared symbolic tail fails to converge. The exact result, retaining the requested order, is (21, [[0, [61/2, 27, 101/4, 121/5, 47/2, 23, 181/8, 67/3, 221/10, 241/11, 87/4, 281/13, 43/2, 107/5, 341/16, 361/17, 127/6, 401/19, 421/20, 21, 461/22]], [1, [10, 41/3, 31/2, 83/5, 52/3, 125/7, 73/4, 167/9, 94/5, 19, 115/6, 251/13, 136/7, 293/15, 157/8, 335/17, 178/9, 377/19, 199/10, 419/21, 20]]], No, finite_prefix_not_pointwise_convergence_certificate).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-prefix-boundary · candidate
97. A Long Prefix Still Does Not Replace the All-n Tail
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[21, 22], coefficients=[22, -23], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=22, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].
View answer and explanation
(22, [[0, [32, 85/3, 53/2, 127/5, 74/3, 169/7, 95/4, 211/9, 116/5, 23, 137/6, 295/13, 158/7, 337/15, 179/8, 379/17, 200/9, 421/19, 221/10, 463/21, 22, 505/23]], [1, [21/2, 43/3, 65/4, 87/5, 109/6, 131/7, 153/8, 175/9, 197/10, 219/11, 241/12, 263/13, 285/14, 307/15, 329/16, 351/17, 373/18, 395/19, 417/20, 439/21, 461/22, 21]]], No, finite_prefix_not_pointwise_convergence_certificate)
Display only the requested finite values. The No sufficiency flag concerns inference from this prefix alone, not a claim that the separately declared symbolic tail fails to converge. The exact result, retaining the requested order, is (22, [[0, [32, 85/3, 53/2, 127/5, 74/3, 169/7, 95/4, 211/9, 116/5, 23, 137/6, 295/13, 158/7, 337/15, 179/8, 379/17, 200/9, 421/19, 221/10, 463/21, 22, 505/23]], [1, [21/2, 43/3, 65/4, 87/5, 109/6, 131/7, 153/8, 175/9, 197/10, 219/11, 241/12, 263/13, 285/14, 307/15, 329/16, 351/17, 373/18, 395/19, 417/20, 439/21, 461/22, 21]]], No, finite_prefix_not_pointwise_convergence_certificate).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-prefix-boundary · candidate
98. A Long Prefix Still Does Not Replace the All-n Tail
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[22, 23], coefficients=[23, -24], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=23, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].
View answer and explanation
(23, [[0, [67/2, 89/3, 111/4, 133/5, 155/6, 177/7, 199/8, 221/9, 243/10, 265/11, 287/12, 309/13, 331/14, 353/15, 375/16, 397/17, 419/18, 441/19, 463/20, 485/21, 507/22, 23, 551/24]], [1, [11, 15, 17, 91/5, 19, 137/7, 20, 61/3, 103/5, 229/11, 21, 275/13, 149/7, 107/5, 43/2, 367/17, 65/3, 413/19, 109/5, 153/7, 241/11, 505/23, 22]]], No, finite_prefix_not_pointwise_convergence_certificate)
Display only the requested finite values. The No sufficiency flag concerns inference from this prefix alone, not a claim that the separately declared symbolic tail fails to converge. The exact result, retaining the requested order, is (23, [[0, [67/2, 89/3, 111/4, 133/5, 155/6, 177/7, 199/8, 221/9, 243/10, 265/11, 287/12, 309/13, 331/14, 353/15, 375/16, 397/17, 419/18, 441/19, 463/20, 485/21, 507/22, 23, 551/24]], [1, [11, 15, 17, 91/5, 19, 137/7, 20, 61/3, 103/5, 229/11, 21, 275/13, 149/7, 107/5, 43/2, 367/17, 65/3, 413/19, 109/5, 153/7, 241/11, 505/23, 22]]], No, finite_prefix_not_pointwise_convergence_certificate).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-prefix-boundary · candidate
99. A Long Prefix Still Does Not Replace the All-n Tail
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[23, 24], coefficients=[24, -25], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=24, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].
View answer and explanation
(24, [[0, [35, 31, 29, 139/5, 27, 185/7, 26, 77/3, 127/5, 277/11, 25, 323/13, 173/7, 123/5, 49/2, 415/17, 73/3, 461/19, 121/5, 169/7, 265/11, 553/23, 24, 599/25]], [1, [23/2, 47/3, 71/4, 19, 119/6, 143/7, 167/8, 191/9, 43/2, 239/11, 263/12, 287/13, 311/14, 67/3, 359/16, 383/17, 407/18, 431/19, 91/4, 479/21, 503/22, 527/23, 551/24, 23]]], No, finite_prefix_not_pointwise_convergence_certificate)
Display only the requested finite values. The No sufficiency flag concerns inference from this prefix alone, not a claim that the separately declared symbolic tail fails to converge. The exact result, retaining the requested order, is (24, [[0, [35, 31, 29, 139/5, 27, 185/7, 26, 77/3, 127/5, 277/11, 25, 323/13, 173/7, 123/5, 49/2, 415/17, 73/3, 461/19, 121/5, 169/7, 265/11, 553/23, 24, 599/25]], [1, [23/2, 47/3, 71/4, 19, 119/6, 143/7, 167/8, 191/9, 43/2, 239/11, 263/12, 287/13, 311/14, 67/3, 359/16, 383/17, 407/18, 431/19, 91/4, 479/21, 503/22, 527/23, 551/24, 23]]], No, finite_prefix_not_pointwise_convergence_certificate).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-prefix-boundary · candidate
100. A Long Prefix Still Does Not Replace the All-n Tail
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[24, 25], coefficients=[25, -26], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=25, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].
View answer and explanation
(25, [[0, [73/2, 97/3, 121/4, 29, 169/6, 193/7, 217/8, 241/9, 53/2, 289/11, 313/12, 337/13, 361/14, 77/3, 409/16, 433/17, 457/18, 481/19, 101/4, 529/21, 553/22, 577/23, 601/24, 25, 649/26]], [1, [12, 49/3, 37/2, 99/5, 62/3, 149/7, 87/4, 199/9, 112/5, 249/11, 137/6, 23, 162/7, 349/15, 187/8, 399/17, 212/9, 449/19, 237/10, 499/21, 262/11, 549/23, 287/12, 599/25, 24]]], No, finite_prefix_not_pointwise_convergence_certificate)
Display only the requested finite values. The No sufficiency flag concerns inference from this prefix alone, not a claim that the separately declared symbolic tail fails to converge. The exact result, retaining the requested order, is (25, [[0, [73/2, 97/3, 121/4, 29, 169/6, 193/7, 217/8, 241/9, 53/2, 289/11, 313/12, 337/13, 361/14, 77/3, 409/16, 433/17, 457/18, 481/19, 101/4, 529/21, 553/22, 577/23, 601/24, 25, 649/26]], [1, [12, 49/3, 37/2, 99/5, 62/3, 149/7, 87/4, 199/9, 112/5, 249/11, 137/6, 23, 162/7, 349/15, 187/8, 399/17, 212/9, 449/19, 237/10, 499/21, 262/11, 549/23, 287/12, 599/25, 24]]], No, finite_prefix_not_pointwise_convergence_certificate).
Reviewed step-by-step derivation
Pointwise convergence of functions · qf-mod34-pointwise-finite-prefix-boundary · candidate
Canonical question JSONL · English JSONL · Validation report