Question 17583: A Long Prefix Still Does Not Replace the All-n Tail

Original question and givens

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].

• G001 — 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].

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-pointwise-convergence-of-functions

Step-by-step derivation

same COMPLETE finite domain Fin(2);original aligned limits=[7, 8];signed coefficients=[8, -9];positive shift=2

declared for EVERY n>=0:f_n(x)=limit[x]+coefficient[x]/(n+shift);signed error=f_n(x)-limit[x]=coefficient[x]/(n+shift)

finite prefix point0 index0 denominator := sum(0, 2)

finite prefix point0 index0 denominator = 2

finite prefix point0 index0 signed correction := div(8, 2)

finite prefix point0 index0 signed correction = 4

finite prefix point0 index0 function value := sum(7, 4)

finite prefix point0 index0 function value = 11

finite prefix point0 index0 reverse signed error := sub(11, 7)

finite prefix point0 index0 reverse signed error = 4

finite prefix point0 index0:original point=0;n=0;limit=7;signed coefficient=8

f_n(x)=11;f_n(x)-limit=4;this error field remains SIGNED

finite prefix point0 index1 denominator := sum(1, 2)

finite prefix point0 index1 denominator = 3

finite prefix point0 index1 signed correction := div(8, 3)

finite prefix point0 index1 signed correction = 8/3

finite prefix point0 index1 function value := sum(7, 8/3)

finite prefix point0 index1 function value = 29/3

finite prefix point0 index1 reverse signed error := sub(29/3, 7)

finite prefix point0 index1 reverse signed error = 8/3

finite prefix point0 index1:original point=0;n=1;limit=7;signed coefficient=8

f_n(x)=29/3;f_n(x)-limit=8/3;this error field remains SIGNED

finite prefix point0 index2 denominator := sum(2, 2)

finite prefix point0 index2 denominator = 4

finite prefix point0 index2 signed correction := div(8, 4)

finite prefix point0 index2 signed correction = 2

finite prefix point0 index2 function value := sum(7, 2)

finite prefix point0 index2 function value = 9

finite prefix point0 index2 reverse signed error := sub(9, 7)

finite prefix point0 index2 reverse signed error = 2

finite prefix point0 index2:original point=0;n=2;limit=7;signed coefficient=8

f_n(x)=9;f_n(x)-limit=2;this error field remains SIGNED

finite prefix point0 index3 denominator := sum(3, 2)

finite prefix point0 index3 denominator = 5

finite prefix point0 index3 signed correction := div(8, 5)

finite prefix point0 index3 signed correction = 8/5

finite prefix point0 index3 function value := sum(7, 8/5)

finite prefix point0 index3 function value = 43/5

finite prefix point0 index3 reverse signed error := sub(43/5, 7)

finite prefix point0 index3 reverse signed error = 8/5

finite prefix point0 index3:original point=0;n=3;limit=7;signed coefficient=8

f_n(x)=43/5;f_n(x)-limit=8/5;this error field remains SIGNED

finite prefix point0 index4 denominator := sum(4, 2)

finite prefix point0 index4 denominator = 6

finite prefix point0 index4 signed correction := div(8, 6)

finite prefix point0 index4 signed correction = 4/3

finite prefix point0 index4 function value := sum(7, 4/3)

finite prefix point0 index4 function value = 25/3

finite prefix point0 index4 reverse signed error := sub(25/3, 7)

finite prefix point0 index4 reverse signed error = 4/3

finite prefix point0 index4:original point=0;n=4;limit=7;signed coefficient=8

f_n(x)=25/3;f_n(x)-limit=4/3;this error field remains SIGNED

finite prefix point0 index5 denominator := sum(5, 2)

finite prefix point0 index5 denominator = 7

finite prefix point0 index5 signed correction := div(8, 7)

finite prefix point0 index5 signed correction = 8/7

finite prefix point0 index5 function value := sum(7, 8/7)

finite prefix point0 index5 function value = 57/7

finite prefix point0 index5 reverse signed error := sub(57/7, 7)

finite prefix point0 index5 reverse signed error = 8/7

finite prefix point0 index5:original point=0;n=5;limit=7;signed coefficient=8

f_n(x)=57/7;f_n(x)-limit=8/7;this error field remains SIGNED

finite prefix point0 index6 denominator := sum(6, 2)

finite prefix point0 index6 denominator = 8

finite prefix point0 index6 signed correction := div(8, 8)

finite prefix point0 index6 signed correction = 1

finite prefix point0 index6 function value := sum(7, 1)

finite prefix point0 index6 function value = 8

finite prefix point0 index6 reverse signed error := sub(8, 7)

finite prefix point0 index6 reverse signed error = 1

finite prefix point0 index6:original point=0;n=6;limit=7;signed coefficient=8

f_n(x)=8;f_n(x)-limit=1;this error field remains SIGNED

finite prefix point0 index7 denominator := sum(7, 2)

finite prefix point0 index7 denominator = 9

finite prefix point0 index7 signed correction := div(8, 9)

finite prefix point0 index7 signed correction = 8/9

finite prefix point0 index7 function value := sum(7, 8/9)

finite prefix point0 index7 function value = 71/9

finite prefix point0 index7 reverse signed error := sub(71/9, 7)

finite prefix point0 index7 reverse signed error = 8/9

finite prefix point0 index7:original point=0;n=7;limit=7;signed coefficient=8

f_n(x)=71/9;f_n(x)-limit=8/9;this error field remains SIGNED

finite prefix point1 index0 denominator := sum(0, 2)

finite prefix point1 index0 denominator = 2

finite prefix point1 index0 signed correction := div(-9, 2)

finite prefix point1 index0 signed correction = -9/2

finite prefix point1 index0 function value := sum(8, -9/2)

finite prefix point1 index0 function value = 7/2

finite prefix point1 index0 reverse signed error := sub(7/2, 8)

finite prefix point1 index0 reverse signed error = -9/2

finite prefix point1 index0:original point=1;n=0;limit=8;signed coefficient=-9

f_n(x)=7/2;f_n(x)-limit=-9/2;this error field remains SIGNED

finite prefix point1 index1 denominator := sum(1, 2)

finite prefix point1 index1 denominator = 3

finite prefix point1 index1 signed correction := div(-9, 3)

finite prefix point1 index1 signed correction = -3

finite prefix point1 index1 function value := sum(8, -3)

finite prefix point1 index1 function value = 5

finite prefix point1 index1 reverse signed error := sub(5, 8)

finite prefix point1 index1 reverse signed error = -3

finite prefix point1 index1:original point=1;n=1;limit=8;signed coefficient=-9

f_n(x)=5;f_n(x)-limit=-3;this error field remains SIGNED

finite prefix point1 index2 denominator := sum(2, 2)

finite prefix point1 index2 denominator = 4

finite prefix point1 index2 signed correction := div(-9, 4)

finite prefix point1 index2 signed correction = -9/4

finite prefix point1 index2 function value := sum(8, -9/4)

finite prefix point1 index2 function value = 23/4

finite prefix point1 index2 reverse signed error := sub(23/4, 8)

finite prefix point1 index2 reverse signed error = -9/4

finite prefix point1 index2:original point=1;n=2;limit=8;signed coefficient=-9

f_n(x)=23/4;f_n(x)-limit=-9/4;this error field remains SIGNED

finite prefix point1 index3 denominator := sum(3, 2)

finite prefix point1 index3 denominator = 5

finite prefix point1 index3 signed correction := div(-9, 5)

finite prefix point1 index3 signed correction = -9/5

finite prefix point1 index3 function value := sum(8, -9/5)

finite prefix point1 index3 function value = 31/5

finite prefix point1 index3 reverse signed error := sub(31/5, 8)

finite prefix point1 index3 reverse signed error = -9/5

finite prefix point1 index3:original point=1;n=3;limit=8;signed coefficient=-9

f_n(x)=31/5;f_n(x)-limit=-9/5;this error field remains SIGNED

finite prefix point1 index4 denominator := sum(4, 2)

finite prefix point1 index4 denominator = 6

finite prefix point1 index4 signed correction := div(-9, 6)

finite prefix point1 index4 signed correction = -3/2

finite prefix point1 index4 function value := sum(8, -3/2)

finite prefix point1 index4 function value = 13/2

finite prefix point1 index4 reverse signed error := sub(13/2, 8)

finite prefix point1 index4 reverse signed error = -3/2

finite prefix point1 index4:original point=1;n=4;limit=8;signed coefficient=-9

f_n(x)=13/2;f_n(x)-limit=-3/2;this error field remains SIGNED

finite prefix point1 index5 denominator := sum(5, 2)

finite prefix point1 index5 denominator = 7

finite prefix point1 index5 signed correction := div(-9, 7)

finite prefix point1 index5 signed correction = -9/7

finite prefix point1 index5 function value := sum(8, -9/7)

finite prefix point1 index5 function value = 47/7

finite prefix point1 index5 reverse signed error := sub(47/7, 8)

finite prefix point1 index5 reverse signed error = -9/7

finite prefix point1 index5:original point=1;n=5;limit=8;signed coefficient=-9

f_n(x)=47/7;f_n(x)-limit=-9/7;this error field remains SIGNED

finite prefix point1 index6 denominator := sum(6, 2)

finite prefix point1 index6 denominator = 8

finite prefix point1 index6 signed correction := div(-9, 8)

finite prefix point1 index6 signed correction = -9/8

finite prefix point1 index6 function value := sum(8, -9/8)

finite prefix point1 index6 function value = 55/8

finite prefix point1 index6 reverse signed error := sub(55/8, 8)

finite prefix point1 index6 reverse signed error = -9/8

finite prefix point1 index6:original point=1;n=6;limit=8;signed coefficient=-9

f_n(x)=55/8;f_n(x)-limit=-9/8;this error field remains SIGNED

finite prefix point1 index7 denominator := sum(7, 2)

finite prefix point1 index7 denominator = 9

finite prefix point1 index7 signed correction := div(-9, 9)

finite prefix point1 index7 signed correction = -1

finite prefix point1 index7 function value := sum(8, -1)

finite prefix point1 index7 function value = 7

finite prefix point1 index7 reverse signed error := sub(7, 8)

finite prefix point1 index7 reverse signed error = -1

finite prefix point1 index7:original point=1;n=7;limit=8;signed coefficient=-9

f_n(x)=7;f_n(x)-limit=-1;this error field remains SIGNED

finite observed length=8;complete domain-order rows=[[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]]]

prefix-alone sufficiency=No;finite_prefix_not_pointwise_convergence_certificate

Reverse checks

• S082 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S080 · premise_and_result_audit — Recompute finite prefix point1 index7 reverse signed error with exact rational arithmetic, not rounded decimals.
• S079 · premise_and_result_audit — Recompute finite prefix point1 index7 function value with exact rational arithmetic, not rounded decimals.
• S078 · premise_and_result_audit — Recompute finite prefix point1 index7 signed correction with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Recompute finite prefix point1 index7 denominator with exact rational arithmetic, not rounded decimals.
• S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S075 · premise_and_result_audit — Recompute finite prefix point1 index6 reverse signed error with exact rational arithmetic, not rounded decimals.
• S074 · premise_and_result_audit — Recompute finite prefix point1 index6 function value with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Recompute finite prefix point1 index6 signed correction with exact rational arithmetic, not rounded decimals.
• S072 · premise_and_result_audit — Recompute finite prefix point1 index6 denominator with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S070 · premise_and_result_audit — Recompute finite prefix point1 index5 reverse signed error with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Recompute finite prefix point1 index5 function value with exact rational arithmetic, not rounded decimals.
• S068 · premise_and_result_audit — Recompute finite prefix point1 index5 signed correction with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Recompute finite prefix point1 index5 denominator with exact rational arithmetic, not rounded decimals.
• S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S065 · premise_and_result_audit — Recompute finite prefix point1 index4 reverse signed error with exact rational arithmetic, not rounded decimals.
• S064 · premise_and_result_audit — Recompute finite prefix point1 index4 function value with exact rational arithmetic, not rounded decimals.
• S063 · premise_and_result_audit — Recompute finite prefix point1 index4 signed correction with exact rational arithmetic, not rounded decimals.
• S062 · premise_and_result_audit — Recompute finite prefix point1 index4 denominator with exact rational arithmetic, not rounded decimals.
• S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S060 · premise_and_result_audit — Recompute finite prefix point1 index3 reverse signed error with exact rational arithmetic, not rounded decimals.
• S059 · premise_and_result_audit — Recompute finite prefix point1 index3 function value with exact rational arithmetic, not rounded decimals.
• S058 · premise_and_result_audit — Recompute finite prefix point1 index3 signed correction with exact rational arithmetic, not rounded decimals.
• S057 · premise_and_result_audit — Recompute finite prefix point1 index3 denominator with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S055 · premise_and_result_audit — Recompute finite prefix point1 index2 reverse signed error with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Recompute finite prefix point1 index2 function value with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute finite prefix point1 index2 signed correction with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute finite prefix point1 index2 denominator with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S050 · premise_and_result_audit — Recompute finite prefix point1 index1 reverse signed error with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute finite prefix point1 index1 function value with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute finite prefix point1 index1 signed correction with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute finite prefix point1 index1 denominator with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S045 · premise_and_result_audit — Recompute finite prefix point1 index0 reverse signed error with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute finite prefix point1 index0 function value with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Recompute finite prefix point1 index0 signed correction with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute finite prefix point1 index0 denominator with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S040 · premise_and_result_audit — Recompute finite prefix point0 index7 reverse signed error with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute finite prefix point0 index7 function value with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute finite prefix point0 index7 signed correction with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute finite prefix point0 index7 denominator with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S035 · premise_and_result_audit — Recompute finite prefix point0 index6 reverse signed error with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute finite prefix point0 index6 function value with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute finite prefix point0 index6 signed correction with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute finite prefix point0 index6 denominator with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S030 · premise_and_result_audit — Recompute finite prefix point0 index5 reverse signed error with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute finite prefix point0 index5 function value with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute finite prefix point0 index5 signed correction with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute finite prefix point0 index5 denominator with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S025 · premise_and_result_audit — Recompute finite prefix point0 index4 reverse signed error with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute finite prefix point0 index4 function value with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute finite prefix point0 index4 signed correction with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute finite prefix point0 index4 denominator with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Recompute finite prefix point0 index3 reverse signed error with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute finite prefix point0 index3 function value with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute finite prefix point0 index3 signed correction with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute finite prefix point0 index3 denominator with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Recompute finite prefix point0 index2 reverse signed error with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute finite prefix point0 index2 function value with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute finite prefix point0 index2 signed correction with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute finite prefix point0 index2 denominator with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Recompute finite prefix point0 index1 reverse signed error with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute finite prefix point0 index1 function value with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute finite prefix point0 index1 signed correction with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute finite prefix point0 index1 denominator with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Recompute finite prefix point0 index0 reverse signed error with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute finite prefix point0 index0 function value with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute finite prefix point0 index0 signed correction with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute finite prefix point0 index0 denominator with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(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)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.