Question 12121: Adjacent Opposite Terms Refute Cauchy Behavior

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=21; arbitrary proposed cutoff N=41. 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=21; arbitrary proposed cutoff N=41. 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

Step-by-step derivation

opposite adjacent tail distance := mul(2, 21)

opposite adjacent tail distance = 42

explicit x_n=(21)*(-1)^n;arbitrary proposed cutoff N=41;epsilon=21

indices=[41, 42];opposite signs give distance=42>=epsilon;counterexample=Yes;Cauchy=No

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

(21, 41, 42, 21, 42, 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.