Question 17580: 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=[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].

• G001 — 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].

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=[4, 5];signed coefficients=[5, -6];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(5, 2)

finite prefix point0 index0 signed correction = 5/2

finite prefix point0 index0 function value := sum(4, 5/2)

finite prefix point0 index0 function value = 13/2

finite prefix point0 index0 reverse signed error := sub(13/2, 4)

finite prefix point0 index0 reverse signed error = 5/2

finite prefix point0 index0:original point=0;n=0;limit=4;signed coefficient=5

f_n(x)=13/2;f_n(x)-limit=5/2;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(5, 3)

finite prefix point0 index1 signed correction = 5/3

finite prefix point0 index1 function value := sum(4, 5/3)

finite prefix point0 index1 function value = 17/3

finite prefix point0 index1 reverse signed error := sub(17/3, 4)

finite prefix point0 index1 reverse signed error = 5/3

finite prefix point0 index1:original point=0;n=1;limit=4;signed coefficient=5

f_n(x)=17/3;f_n(x)-limit=5/3;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(5, 4)

finite prefix point0 index2 signed correction = 5/4

finite prefix point0 index2 function value := sum(4, 5/4)

finite prefix point0 index2 function value = 21/4

finite prefix point0 index2 reverse signed error := sub(21/4, 4)

finite prefix point0 index2 reverse signed error = 5/4

finite prefix point0 index2:original point=0;n=2;limit=4;signed coefficient=5

f_n(x)=21/4;f_n(x)-limit=5/4;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(5, 5)

finite prefix point0 index3 signed correction = 1

finite prefix point0 index3 function value := sum(4, 1)

finite prefix point0 index3 function value = 5

finite prefix point0 index3 reverse signed error := sub(5, 4)

finite prefix point0 index3 reverse signed error = 1

finite prefix point0 index3:original point=0;n=3;limit=4;signed coefficient=5

f_n(x)=5;f_n(x)-limit=1;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(5, 6)

finite prefix point0 index4 signed correction = 5/6

finite prefix point0 index4 function value := sum(4, 5/6)

finite prefix point0 index4 function value = 29/6

finite prefix point0 index4 reverse signed error := sub(29/6, 4)

finite prefix point0 index4 reverse signed error = 5/6

finite prefix point0 index4:original point=0;n=4;limit=4;signed coefficient=5

f_n(x)=29/6;f_n(x)-limit=5/6;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(-6, 2)

finite prefix point1 index0 signed correction = -3

finite prefix point1 index0 function value := sum(5, -3)

finite prefix point1 index0 function value = 2

finite prefix point1 index0 reverse signed error := sub(2, 5)

finite prefix point1 index0 reverse signed error = -3

finite prefix point1 index0:original point=1;n=0;limit=5;signed coefficient=-6

f_n(x)=2;f_n(x)-limit=-3;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(-6, 3)

finite prefix point1 index1 signed correction = -2

finite prefix point1 index1 function value := sum(5, -2)

finite prefix point1 index1 function value = 3

finite prefix point1 index1 reverse signed error := sub(3, 5)

finite prefix point1 index1 reverse signed error = -2

finite prefix point1 index1:original point=1;n=1;limit=5;signed coefficient=-6

f_n(x)=3;f_n(x)-limit=-2;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(-6, 4)

finite prefix point1 index2 signed correction = -3/2

finite prefix point1 index2 function value := sum(5, -3/2)

finite prefix point1 index2 function value = 7/2

finite prefix point1 index2 reverse signed error := sub(7/2, 5)

finite prefix point1 index2 reverse signed error = -3/2

finite prefix point1 index2:original point=1;n=2;limit=5;signed coefficient=-6

f_n(x)=7/2;f_n(x)-limit=-3/2;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(-6, 5)

finite prefix point1 index3 signed correction = -6/5

finite prefix point1 index3 function value := sum(5, -6/5)

finite prefix point1 index3 function value = 19/5

finite prefix point1 index3 reverse signed error := sub(19/5, 5)

finite prefix point1 index3 reverse signed error = -6/5

finite prefix point1 index3:original point=1;n=3;limit=5;signed coefficient=-6

f_n(x)=19/5;f_n(x)-limit=-6/5;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(-6, 6)

finite prefix point1 index4 signed correction = -1

finite prefix point1 index4 function value := sum(5, -1)

finite prefix point1 index4 function value = 4

finite prefix point1 index4 reverse signed error := sub(4, 5)

finite prefix point1 index4 reverse signed error = -1

finite prefix point1 index4:original point=1;n=4;limit=5;signed coefficient=-6

f_n(x)=4;f_n(x)-limit=-1;this error field remains SIGNED

finite observed length=5;complete domain-order rows=[[0, [13/2, 17/3, 21/4, 5, 29/6]], [1, [2, 3, 7/2, 19/5, 4]]]

prefix-alone sufficiency=No;finite_prefix_not_pointwise_convergence_certificate

Reverse checks

• S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 index4 reverse signed error with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute finite prefix point1 index4 function value with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute finite prefix point1 index4 signed correction with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute finite prefix point1 index4 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 index3 reverse signed error with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute finite prefix point1 index3 function value with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Recompute finite prefix point1 index3 signed correction with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute finite prefix point1 index3 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 index2 reverse signed error with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute finite prefix point1 index2 function value with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute finite prefix point1 index2 signed correction with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute finite prefix point1 index2 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 point1 index1 reverse signed error with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute finite prefix point1 index1 function value with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute finite prefix point1 index1 signed correction with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute finite prefix point1 index1 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 point1 index0 reverse signed error with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute finite prefix point1 index0 function value with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute finite prefix point1 index0 signed correction with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute finite prefix point1 index0 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

(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)

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.