# Question 28716: Strict Eligibility of a Candidate Tail Cutoff

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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.

R0056-28716 · q-20261007-advance-20261008-r01-sequence-limit-0066



## Original question and givens

For an arbitrary fixed real L, u(n)=L+(-16)/(n+4) on natural indices starting at zero, with d=4≥1. With ε=1/1>0 and candidate N=13, does every n≥N have |u(n)−L|<ε, including n=N?


- G001 — For an arbitrary fixed real L, u(n)=L+(-16)/(n+4) on natural indices starting at zero, with d=4≥1. With ε=1/1>0 and candidate N=13, does every n≥N have |u(n)−L|<ε, including n=N?





## 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-cutoff-validity, cutoff_validity]

After

derived = [strict_all_n_ge_N_tail_validity]

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 = [-16, 4, naturals_from_zero]

After

derived = [16, 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(16, 1)

After

Sequence ORIGINAL absolute c times epsilon denominator = 16

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 = [16, 1]

After

derived = [16, 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.

Step identifiers and dependenciesRule: Sequence_EXACT_floor_quotient_remainder_NO_float_or_ceil · Uses: G001, S003





S005 · calculationBefore

Sequence cutoff ORIGINAL N plus d := sum(13, 4)

After

Sequence cutoff ORIGINAL N plus d = 17

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 cutoff ORIGINAL epsilon numerator times N plus d := mul(1, 17)

After

Sequence cutoff ORIGINAL epsilon numerator times N plus d = 17

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 = [-16, 4, 1, 1, 13, 16, 17]

After

derived = [Yes, Yes]

Conditions: none

Test STRICT < at N;equality is No.Since denominators increase for n>=N,success at N gives the whole tail,and failure at N itself is a counterexample.Do not test N+1 instead,or infer minimality from mere validity.

Step identifiers and dependenciesRule: Sequence_cutoff_STRICT_cross_product_at_N_IFF_ALL_tail · 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 cutoff ORIGINAL epsilon numerator times N plus d with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Recompute Sequence cutoff 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

Yes

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: d588507685415b3652f79d0db557f14fb1594ce983a9983986cb1715ec1e7f2e
- candidate record SHA-256: 8d767fa3f84ddbc7c80c3d8f2ed212d6af1c35905566782746e547dcce8bc766
- 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-0066.json)