Question 12178: An Explicit Infinite Constant Tail Supplies a Cauchy Witness

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 real sequence has finite prefix=[-10/1, 3/2, -1/3] followed forever by tail=-3/3. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

• 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 real sequence has finite prefix=[-10/1, 3/2, -1/3] followed forever by tail=-3/3. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.

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

prefix=[-10, 3/2, -1/3];all infinite later terms explicitly=-1

for every epsilon>0 choose N=3;all m,n>=N have distance0<epsilon

Reverse checks

• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([-10, 3/2, -1/3], -1, 3, for any epsilon>0 choose N=prefix length, 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.