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R0056-28743 · q-20261007-advance-20261008-r01-sequence-limit-0043
For an arbitrary fixed real L, u(n)=L+(-18)/(n+2) on natural indices starting at zero, with d=2≥1. For ε=3/3>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+(-18)/(n+2) on natural indices starting at zero, with d=2≥1. For ε=3/3>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.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.
Before
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.
Rule: R56_source_answer_role_and_EXACT_family_interface · Uses: G001
Before
source roles = [-18, 2, naturals_from_zero]
After
derived = [18, 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.
Rule: Sequence_WHOLE_law_and_positive_increasing_denominator · Uses: G001, S001
Before
Sequence ORIGINAL absolute c times epsilon denominator := mul(18, 3)
After
Sequence ORIGINAL absolute c times epsilon denominator = 54
Conditions: none
Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
source roles = [54, 3]
After
derived = [18, 0, left=quot*eps_n+rem, 0<=rem<eps_n]
Conditions: none
Because eps_n>0,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.
Rule: Sequence_EXACT_floor_quotient_remainder_NO_float_or_ceil · Uses: G001, S003
Before
Sequence least ORIGINAL N plus d := sum(17, 2)
After
Sequence least ORIGINAL N plus d = 19
Conditions: none
Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
Sequence least ORIGINAL epsilon numerator times N plus d := mul(3, 19)
After
Sequence least ORIGINAL epsilon numerator times N plus d = 57
Conditions: none
Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
source roles = [-18, 2, 3, 3, 54, 18]
After
derived = [17, 57, 54, Yes]
Conditions: none
The tail INCLUDES N.Monotonic positive denominators make every n>=N qualify IFF |c|*ed<en*(N+d).The least integer is max(0,floor(|c|*ed/en)-d+1).Only when N>0 test failure at N-1;do not invent a negative index for N=0.Derived N is not SAFE-capped.
Rule: Sequence_LEAST_N_strict_tail_INCLUDING_N_AND_previous_failure · Uses: G001, S006
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.17
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: c6d26affd93c79e0c160e9952e12a9a0a095a2e6cfa459f71d97fa705e5e2ce3candidate record SHA-256: 4a5f28bd3fa194b667e97144a7747d7354281933d44b6c826a3db4f717963ae6semantic review receipt SHA-256: 8bbda4edbed7078634ae88342998c79bcfb95de96187fac119c3b08e0adb0bc8question-first witness SHA-256: 3edafa72d63d16a22414892b55806430e0a25b15b926d04f312550179607759fLean: NOT_RUN · human review: NOT_PERFORMED