# Question 28734: Least Natural Cutoff Including the Cutoff Term [Return to catalogue page 144](https://ocme.evemisslab.com/en/questions/page/144/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261007-advance-20261008-r01-sequence-limit-foundations/#q-20261007-advance-20261008-r01-sequence-limit-0034) 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. R0056-28734 · q-20261007-advance-20261008-r01-sequence-limit-0034 ## Original question and givens For an arbitrary fixed real L, u(n)=L+(9)/(n+1) on natural indices starting at zero, with d=1≥1. For ε=4/3>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε. - G001 — For an arbitrary fixed real L, u(n)=L+(9)/(n+1) on natural indices starting at zero, with d=1≥1. For ε=4/3>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε. ## 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-sequence-limit](https://ocme.evemisslab.com/en/mko/mko-sequence-limit/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-sequence-limit-least-cutoff, least_cutoff] After derived = [least_nonnegative_strict_tail_cutoff] Conditions: none Use the declared scalar/set/Yes-No/fraction answer role;integer fractions retain/1.Do not add an undeclared domain or change original bilingual data,formulas or questions. Step identifiers and dependenciesRule: R56_source_answer_role_and_EXACT_family_interface · Uses: G001 S002 · impliesBefore source roles = [9, 1, naturals_from_zero] After derived = [9, u(n)=L+c/(n+d), |u(n)-L|=|c|/(n+d)] Conditions: none The premise is the WHOLE law for every n>=0,not a finite prefix.With d>0,the denominator is positive and increasing.Subtract L before taking the absolute value.In error/cutoff tasks L is an arbitrary fixed real,not an invented input parameter. Step identifiers and dependenciesRule: Sequence_WHOLE_law_and_positive_increasing_denominator · Uses: G001, S001 S003 · calculationBefore Sequence ORIGINAL absolute c times epsilon denominator := mul(9, 3) After Sequence ORIGINAL absolute c times epsilon denominator = 27 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore source roles = [27, 4] After derived = [6, 3, left=quot*eps_n+rem, 0<=rem0,integer division yields the exact floor and remainder.No floating rounding or ceil replacement is allowed.Derived products/quotients may exceed SAFE;the raw inputs alone are SAFE-bounded. Step identifiers and dependenciesRule: Sequence_EXACT_floor_quotient_remainder_NO_float_or_ceil · Uses: G001, S003 S005 · calculationBefore Sequence least ORIGINAL N plus d := sum(6, 1) After Sequence least ORIGINAL N plus d = 7 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Sequence least ORIGINAL epsilon numerator times N plus d := mul(4, 7) After Sequence least ORIGINAL epsilon numerator times N plus d = 28 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore source roles = [9, 1, 4, 3, 27, 6] After derived = [6, 28, 24, Yes] Conditions: none The tail INCLUDES N.Monotonic positive denominators make every n>=N qualify IFF |c|*ed0 test failure at N-1;do not invent a negative index for N=0.Derived N is not SAFE-capped. Step identifiers and dependenciesRule: Sequence_LEAST_N_strict_tail_INCLUDING_N_AND_previous_failure · Uses: G001, S006 ## Reverse checks - S007 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance. - S006 · premise_and_result_audit — Recompute Sequence least ORIGINAL epsilon numerator times N plus d with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Sequence least ORIGINAL N plus d with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance. - S003 · premise_and_result_audit — Recompute Sequence ORIGINAL absolute c times epsilon denominator with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance. - S001 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance. ## Result 6 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: 5fb2efb9bd6e41fe571fe62cbbe8cd929cc5c13e0eb846e68432b9eeccea4d84 - candidate record SHA-256: 26ee06a27e8731a357dbd3dbf1aa9bebc8fe888ef34370b8166fc966843a0f6c - semantic review receipt SHA-256: 8bbda4edbed7078634ae88342998c79bcfb95de96187fac119c3b08e0adb0bc8 - question-first witness SHA-256: 3edafa72d63d16a22414892b55806430e0a25b15b926d04f312550179607759f - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261008-r01-sequence-limit-0034.json)