# Question 27044: One Positive Index Must Give Strict Error Control for Every Point [Return to catalogue page 136](https://ocme.evemisslab.com/en/questions/page/136/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations/#q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0044) 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. R0053-27044 · q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0044 ## Original question and givens Use exact E=19>0 and epsilon=4/10>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate). - G001 — Use exact E=19>0 and epsilon=4/10>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate). ## 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-uniform-limit-continuity-theorem](https://ocme.evemisslab.com/en/mko/mko-uniform-limit-continuity-theorem/) ## Step-by-step derivation S001 · impliesBefore source roles = [uniform-epsilon-index] After derived = [[error_scale, epsilon, N, error_at_N, strict_error_bound, certificate]] Conditions: none List the complete source field order and mode-specific roles without JavaScript or answer-shape inference;No,undefined and infinity retain their distinct meanings. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · calculationBefore Uniform ORIGINAL error divided by epsilon := div(19, 2/5) After Uniform ORIGINAL error divided by epsilon = 95/2 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Uniform ORIGINAL error at FIRST positive N := div(19, 48) After Uniform ORIGINAL error at FIRST positive N = 19/48 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Uniform previous positive index error := div(19, 47) After Uniform previous positive index error = 19/47 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore source roles = [19, 2/5, 95/2, 47] After derived = [19/48, 19/47, Yes, Yes] Conditions: none E/(n+1)E/epsilon.The positive integer N=max(1,floor(E/epsilon)) makes N+1 strictly larger,including exact-division cases.If N>1,the previous positive index has error E/N>=epsilon,so N is minimal.For every n>=N the sup error decreases;the same N controls all x.No ceil or N=0 substitution. Step identifiers and dependenciesRule: Uniform_FIRST_positive_N_strict_all_x_all_later_n · Uses: G001, S004 ## Reverse checks - S005 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct. - S004 · premise_and_result_audit — Recompute Uniform previous positive index error with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Uniform ORIGINAL error at FIRST positive N with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Uniform ORIGINAL error divided by epsilon with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct. ## Result (19, 2/5, 47, 19/48, Yes, same_N_controls_every_x_in_[0,1]) 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: a035b689001118da5697e0ea9aa64749a2dc4e00a48e11380504482ddce827c3 - candidate record SHA-256: 4be100610b9e24a59e4289a37739193e0428f73a4bd8d394c04173d2977afeea - semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62 - question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0044.json)