Question 17578: 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=[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].

• G001 — 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].

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=[2, 3];signed coefficients=[3, -4];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(3, 2)

finite prefix point0 index0 signed correction = 3/2

finite prefix point0 index0 function value := sum(2, 3/2)

finite prefix point0 index0 function value = 7/2

finite prefix point0 index0 reverse signed error := sub(7/2, 2)

finite prefix point0 index0 reverse signed error = 3/2

finite prefix point0 index0:original point=0;n=0;limit=2;signed coefficient=3

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

finite prefix point0 index1 signed correction = 1

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

finite prefix point0 index1 function value = 3

finite prefix point0 index1 reverse signed error := sub(3, 2)

finite prefix point0 index1 reverse signed error = 1

finite prefix point0 index1:original point=0;n=1;limit=2;signed coefficient=3

f_n(x)=3;f_n(x)-limit=1;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(3, 4)

finite prefix point0 index2 signed correction = 3/4

finite prefix point0 index2 function value := sum(2, 3/4)

finite prefix point0 index2 function value = 11/4

finite prefix point0 index2 reverse signed error := sub(11/4, 2)

finite prefix point0 index2 reverse signed error = 3/4

finite prefix point0 index2:original point=0;n=2;limit=2;signed coefficient=3

f_n(x)=11/4;f_n(x)-limit=3/4;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(-4, 2)

finite prefix point1 index0 signed correction = -2

finite prefix point1 index0 function value := sum(3, -2)

finite prefix point1 index0 function value = 1

finite prefix point1 index0 reverse signed error := sub(1, 3)

finite prefix point1 index0 reverse signed error = -2

finite prefix point1 index0:original point=1;n=0;limit=3;signed coefficient=-4

f_n(x)=1;f_n(x)-limit=-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(-4, 3)

finite prefix point1 index1 signed correction = -4/3

finite prefix point1 index1 function value := sum(3, -4/3)

finite prefix point1 index1 function value = 5/3

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

finite prefix point1 index1 reverse signed error = -4/3

finite prefix point1 index1:original point=1;n=1;limit=3;signed coefficient=-4

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

finite prefix point1 index2 signed correction = -1

finite prefix point1 index2 function value := sum(3, -1)

finite prefix point1 index2 function value = 2

finite prefix point1 index2 reverse signed error := sub(2, 3)

finite prefix point1 index2 reverse signed error = -1

finite prefix point1 index2:original point=1;n=2;limit=3;signed coefficient=-4

f_n(x)=2;f_n(x)-limit=-1;this error field remains SIGNED

finite observed length=3;complete domain-order rows=[[0, [7/2, 3, 11/4]], [1, [1, 5/3, 2]]]

prefix-alone sufficiency=No;finite_prefix_not_pointwise_convergence_certificate

Reverse checks

• S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 index2 reverse signed error with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute finite prefix point1 index2 function value with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute finite prefix point1 index2 signed correction with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute finite prefix point1 index2 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 point1 index1 reverse signed error with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute finite prefix point1 index1 function value with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute finite prefix point1 index1 signed correction with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute finite prefix point1 index1 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 point1 index0 reverse signed error with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute finite prefix point1 index0 function value with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute finite prefix point1 index0 signed correction with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute finite prefix point1 index0 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

(3, [[0, [7/2, 3, 11/4]], [1, [1, 5/3, 2]]], 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.