# Question 17535: Keep the Signed Function Error Separate from Its Absolute Bound [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-0010) 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. R0034-17535 · q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0010 ## Original question and givens Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[9, 10], coefficients=[10, -11], 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. n=10, point=1. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed. - G001 — Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[9, 10], coefficients=[10, -11], 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. n=10, point=1. Give (n,point,f_n_x,limit_x,error), where error=f_n(x)−limit[x] is signed. ## 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=[9, 10];signed coefficients=[10, -11];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 selected function evaluation denominator := sum(10, 2) After selected function evaluation denominator = 12 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 selected function evaluation signed correction := div(-11, 12) After selected function evaluation signed correction = -11/12 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 selected function evaluation function value := sum(10, -11/12) After selected function evaluation function value = 109/12 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 selected function evaluation reverse signed error := sub(109/12, 10) After selected function evaluation reverse signed error = -11/12 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 selected function evaluation:original point=1;n=10;limit=10;signed coefficient=-11 After f_n(x)=109/12;f_n(x)-limit=-11/12;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 ## Reverse checks - 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 selected function evaluation reverse signed error with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute selected function evaluation function value with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute selected function evaluation signed correction with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute selected function evaluation 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 (10, 1, 109/12, 10, -11/12) 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: 9df40b6cb829b8784a0984b10143f771a1401f03dcce2d8a53636738cc6c2f1b - candidate record SHA-256: 8f93eee02f8d4f69114adc33533d5fe14115687162380a28c653bc37778934ac - 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-0010.json)