# Question 17576: A Long Prefix Still Does Not Replace the All-n Tail [Return to catalogue page 88](https://ocme.evemisslab.com/en/questions/page/88/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations/#q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0076) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0034-17576 · q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0076 ## Original question and givens 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]. - G001 — 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]. ## 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](https://ocme.evemisslab.com/en/mko/mko-pointwise-convergence-of-functions/) ## Step-by-step derivation S001 · impliesBefore same COMPLETE finite domain Fin(2);original aligned limits=[0, 1];signed coefficients=[1, -2];positive shift=2 After 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) Conditions: none The complete common finite domain and all-n formula are declared premises,not extrapolated from samples.Function values and errors are not probabilities;they may be signed or exceed1.Do not clip or shift indices. Step identifiers and dependenciesRule: pointwise_explicit_common_domain_all_n_tail_not_probability · Uses: G001 S002 · calculationBefore finite prefix point0 index0 denominator := sum(0, 2) After finite prefix point0 index0 denominator = 2 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore finite prefix point0 index0 signed correction := div(1, 2) After finite prefix point0 index0 signed correction = 1/2 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore finite prefix point0 index0 function value := sum(0, 1/2) After finite prefix point0 index0 function value = 1/2 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore finite prefix point0 index0 reverse signed error := sub(1/2, 0) After finite prefix point0 index0 reverse signed error = 1/2 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore finite prefix point0 index0:original point=0;n=0;limit=0;signed coefficient=1 After f_n(x)=1/2;f_n(x)-limit=1/2;this error field remains SIGNED Conditions: none Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value. Step identifiers and dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S005 S007 · calculationBefore finite prefix point1 index0 denominator := sum(0, 2) After finite prefix point1 index0 denominator = 2 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore finite prefix point1 index0 signed correction := div(-2, 2) After finite prefix point1 index0 signed correction = -1 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore finite prefix point1 index0 function value := sum(1, -1) After finite prefix point1 index0 function value = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore finite prefix point1 index0 reverse signed error := sub(0, 1) After finite prefix point1 index0 reverse signed error = -1 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore finite prefix point1 index0:original point=1;n=0;limit=1;signed coefficient=-2 After f_n(x)=0;f_n(x)-limit=-1;this error field remains SIGNED Conditions: none Add the correction to the limit,then subtract the limit to reverse-check the error.Keep the evaluate field signed;only the epsilon bound uses absolute value. Step identifiers and dependenciesRule: pointwise_selected_value_reverse_error_not_absolute_value · Uses: G001, S010 S012 · impliesBefore finite observed length=1;complete domain-order rows=[[0, [1/2]], [1, [0]]] After prefix-alone sufficiency=No;finite_prefix_not_pointwise_convergence_certificate Conditions: none This No limits inference from the finite prefix alone,not the separate declared all-n formula.A pointwise witness may depend on x and epsilon and must cover every later n. Step identifiers and dependenciesRule: pointwise_finite_prefix_no_not_declared_model_failure · Uses: G001, S011 ## Reverse checks - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 point1 index0 reverse signed error with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute finite prefix point1 index0 function value with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute finite prefix point1 index0 signed correction with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute finite prefix point1 index0 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 (1, [[0, [1/2]], [1, [0]]], 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 913f6751a6cb6966b33ebb375c0c622cb191de065ef7b2277a81c6e7ceb204f4 - candidate record SHA-256: 508265099afbb24a52f9819fb9ba98942ce0485426ec48d5fc6103d201669bac - semantic review receipt SHA-256: f73d099ebe3ff1c69a5d96f08c75b11b9e5b05120af40cf3ef388fce3b5c5336 - question-first witness SHA-256: 22675ffb6209b4fea076032b29b96e3dc19aad4244c54493f5f65f5e5d875699 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0076.json)