# Question 12111: Adjacent Opposite Terms Refute Cauchy Behavior

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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.

R0023-12111 · q-20261013-advance-20261026-r03-cauchy-sequence-0086



## Original question and givens

Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)<epsilon. No limit or completeness premise is assumed. A finite recorded tail alone does not prove this infinite statement. The infinite sequence alternates between amplitude and its negative, with positive amplitude=11; arbitrary proposed cutoff N=21. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.


- G001 — Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)<epsilon. No limit or completeness premise is assumed. A finite recorded tail alone does not prove this infinite statement. The infinite sequence alternates between amplitude and its negative, with positive amplitude=11; arbitrary proposed cutoff N=21. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.





## 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-cauchy-sequence](https://ocme.evemisslab.com/en/mko/mko-cauchy-sequence/)





## Step-by-step derivation

S001 · calculationBefore

opposite adjacent tail distance := mul(2, 11)

After

opposite adjacent tail distance = 22

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001





S002 · impliesBefore

explicit x_n=(11)*(-1)^n;arbitrary proposed cutoff N=21;epsilon=11

After

indices=[21, 22];opposite signs give distance=22>=epsilon;counterexample=Yes;Cauchy=No

Conditions: none

Fix positive epsilon=A.For any cutoff use N and N+1,both in the tail with opposite signs and distance2A≥A.Thus every cutoff fails.The last flag is Cauchy,not not_cauchy,following the corrected source protocol.

Step identifiers and dependenciesRule: alternating_fixed_epsilon_every_cutoff_counterexample · Uses: G001, S001









## Reverse checks


- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Recompute opposite adjacent tail distance with exact rational arithmetic, not rounded decimals.





## Result

(11, 21, 22, 11, 22, Yes, No)

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: 0df6ebfcadabe52b4adc59a3b6264ed6c063babaa37accbbf4a45f84606a6911
- candidate record SHA-256: 72657a3cbbd9599d7443a9dcb3a802baa86c907eb0d44ce0c12e359131dbbdeb
- semantic review receipt SHA-256: f783e81f44ff0faa13dbf948657e9187244a59a809ebdf96f22cca24966649ff
- question-first witness SHA-256: 49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0086.json)