Question 17582: 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=[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].

• G001 — 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].

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=[6, 7];signed coefficients=[7, -8];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(7, 2)

finite prefix point0 index0 signed correction = 7/2

finite prefix point0 index0 function value := sum(6, 7/2)

finite prefix point0 index0 function value = 19/2

finite prefix point0 index0 reverse signed error := sub(19/2, 6)

finite prefix point0 index0 reverse signed error = 7/2

finite prefix point0 index0:original point=0;n=0;limit=6;signed coefficient=7

f_n(x)=19/2;f_n(x)-limit=7/2;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(7, 3)

finite prefix point0 index1 signed correction = 7/3

finite prefix point0 index1 function value := sum(6, 7/3)

finite prefix point0 index1 function value = 25/3

finite prefix point0 index1 reverse signed error := sub(25/3, 6)

finite prefix point0 index1 reverse signed error = 7/3

finite prefix point0 index1:original point=0;n=1;limit=6;signed coefficient=7

f_n(x)=25/3;f_n(x)-limit=7/3;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(7, 4)

finite prefix point0 index2 signed correction = 7/4

finite prefix point0 index2 function value := sum(6, 7/4)

finite prefix point0 index2 function value = 31/4

finite prefix point0 index2 reverse signed error := sub(31/4, 6)

finite prefix point0 index2 reverse signed error = 7/4

finite prefix point0 index2:original point=0;n=2;limit=6;signed coefficient=7

f_n(x)=31/4;f_n(x)-limit=7/4;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(7, 5)

finite prefix point0 index3 signed correction = 7/5

finite prefix point0 index3 function value := sum(6, 7/5)

finite prefix point0 index3 function value = 37/5

finite prefix point0 index3 reverse signed error := sub(37/5, 6)

finite prefix point0 index3 reverse signed error = 7/5

finite prefix point0 index3:original point=0;n=3;limit=6;signed coefficient=7

f_n(x)=37/5;f_n(x)-limit=7/5;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(7, 6)

finite prefix point0 index4 signed correction = 7/6

finite prefix point0 index4 function value := sum(6, 7/6)

finite prefix point0 index4 function value = 43/6

finite prefix point0 index4 reverse signed error := sub(43/6, 6)

finite prefix point0 index4 reverse signed error = 7/6

finite prefix point0 index4:original point=0;n=4;limit=6;signed coefficient=7

f_n(x)=43/6;f_n(x)-limit=7/6;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(7, 7)

finite prefix point0 index5 signed correction = 1

finite prefix point0 index5 function value := sum(6, 1)

finite prefix point0 index5 function value = 7

finite prefix point0 index5 reverse signed error := sub(7, 6)

finite prefix point0 index5 reverse signed error = 1

finite prefix point0 index5:original point=0;n=5;limit=6;signed coefficient=7

f_n(x)=7;f_n(x)-limit=1;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(7, 8)

finite prefix point0 index6 signed correction = 7/8

finite prefix point0 index6 function value := sum(6, 7/8)

finite prefix point0 index6 function value = 55/8

finite prefix point0 index6 reverse signed error := sub(55/8, 6)

finite prefix point0 index6 reverse signed error = 7/8

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

f_n(x)=55/8;f_n(x)-limit=7/8;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(-8, 2)

finite prefix point1 index0 signed correction = -4

finite prefix point1 index0 function value := sum(7, -4)

finite prefix point1 index0 function value = 3

finite prefix point1 index0 reverse signed error := sub(3, 7)

finite prefix point1 index0 reverse signed error = -4

finite prefix point1 index0:original point=1;n=0;limit=7;signed coefficient=-8

f_n(x)=3;f_n(x)-limit=-4;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(-8, 3)

finite prefix point1 index1 signed correction = -8/3

finite prefix point1 index1 function value := sum(7, -8/3)

finite prefix point1 index1 function value = 13/3

finite prefix point1 index1 reverse signed error := sub(13/3, 7)

finite prefix point1 index1 reverse signed error = -8/3

finite prefix point1 index1:original point=1;n=1;limit=7;signed coefficient=-8

f_n(x)=13/3;f_n(x)-limit=-8/3;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(-8, 4)

finite prefix point1 index2 signed correction = -2

finite prefix point1 index2 function value := sum(7, -2)

finite prefix point1 index2 function value = 5

finite prefix point1 index2 reverse signed error := sub(5, 7)

finite prefix point1 index2 reverse signed error = -2

finite prefix point1 index2:original point=1;n=2;limit=7;signed coefficient=-8

f_n(x)=5;f_n(x)-limit=-2;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(-8, 5)

finite prefix point1 index3 signed correction = -8/5

finite prefix point1 index3 function value := sum(7, -8/5)

finite prefix point1 index3 function value = 27/5

finite prefix point1 index3 reverse signed error := sub(27/5, 7)

finite prefix point1 index3 reverse signed error = -8/5

finite prefix point1 index3:original point=1;n=3;limit=7;signed coefficient=-8

f_n(x)=27/5;f_n(x)-limit=-8/5;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(-8, 6)

finite prefix point1 index4 signed correction = -4/3

finite prefix point1 index4 function value := sum(7, -4/3)

finite prefix point1 index4 function value = 17/3

finite prefix point1 index4 reverse signed error := sub(17/3, 7)

finite prefix point1 index4 reverse signed error = -4/3

finite prefix point1 index4:original point=1;n=4;limit=7;signed coefficient=-8

f_n(x)=17/3;f_n(x)-limit=-4/3;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(-8, 7)

finite prefix point1 index5 signed correction = -8/7

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

finite prefix point1 index5 function value = 41/7

finite prefix point1 index5 reverse signed error := sub(41/7, 7)

finite prefix point1 index5 reverse signed error = -8/7

finite prefix point1 index5:original point=1;n=5;limit=7;signed coefficient=-8

f_n(x)=41/7;f_n(x)-limit=-8/7;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(-8, 8)

finite prefix point1 index6 signed correction = -1

finite prefix point1 index6 function value := sum(7, -1)

finite prefix point1 index6 function value = 6

finite prefix point1 index6 reverse signed error := sub(6, 7)

finite prefix point1 index6 reverse signed error = -1

finite prefix point1 index6:original point=1;n=6;limit=7;signed coefficient=-8

f_n(x)=6;f_n(x)-limit=-1;this error field remains SIGNED

finite observed length=7;complete domain-order rows=[[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]]]

prefix-alone sufficiency=No;finite_prefix_not_pointwise_convergence_certificate

Reverse checks

• S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 index6 reverse signed error with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Recompute finite prefix point1 index6 function value with exact rational arithmetic, not rounded decimals.
• S068 · premise_and_result_audit — Recompute finite prefix point1 index6 signed correction with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Recompute finite prefix point1 index6 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 index5 reverse signed error with exact rational arithmetic, not rounded decimals.
• S064 · premise_and_result_audit — Recompute finite prefix point1 index5 function value with exact rational arithmetic, not rounded decimals.
• S063 · premise_and_result_audit — Recompute finite prefix point1 index5 signed correction with exact rational arithmetic, not rounded decimals.
• S062 · premise_and_result_audit — Recompute finite prefix point1 index5 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 index4 reverse signed error with exact rational arithmetic, not rounded decimals.
• S059 · premise_and_result_audit — Recompute finite prefix point1 index4 function value with exact rational arithmetic, not rounded decimals.
• S058 · premise_and_result_audit — Recompute finite prefix point1 index4 signed correction with exact rational arithmetic, not rounded decimals.
• S057 · premise_and_result_audit — Recompute finite prefix point1 index4 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 index3 reverse signed error with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Recompute finite prefix point1 index3 function value with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute finite prefix point1 index3 signed correction with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute finite prefix point1 index3 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 index2 reverse signed error with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute finite prefix point1 index2 function value with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute finite prefix point1 index2 signed correction with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute finite prefix point1 index2 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 index1 reverse signed error with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute finite prefix point1 index1 function value with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Recompute finite prefix point1 index1 signed correction with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute finite prefix point1 index1 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 point1 index0 reverse signed error with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute finite prefix point1 index0 function value with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute finite prefix point1 index0 signed correction with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute finite prefix point1 index0 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

(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)

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.