Question 12128: A Finite Tail Audit Cannot Establish General 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. Recorded rational sequence=[1/4, 1/5, 1/6, 1/7, 1/8], epsilon=1/5, cutoff N=2; only listed zero-based indices are available. Give (normalized sequence,epsilon,N,all bad [m,n,distance] records in ordered-index order,finite_tail_pass?,general Cauchy established by this audit?,finite-tail-insufficient-for-general-Cauchy). A bad distance is at least epsilon. 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. Recorded rational sequence=[1/4, 1/5, 1/6, 1/7, 1/8], epsilon=1/5, cutoff N=2; only listed zero-based indices are available. Give (normalized sequence,epsilon,N,all bad [m,n,distance] records in ordered-index order,finite_tail_pass?,general Cauchy established by this audit?,finite-tail-insufficient-for-general-Cauchy). A bad distance is at least epsilon. 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

raw sequence=[[1, 4], [1, 5], [1, 6], [1, 7], [1, 8]];raw epsilon=[1, 5]

normalized sequence=[1/4, 1/5, 1/6, 1/7, 1/8];positive epsilon=1/5;cutoff=2

recorded pair2,2 signed difference := sub(1/6, 1/6)

recorded pair2,2 signed difference = 0

ordered recorded indices=[2, 2];signed difference=0;epsilon=1/5

absolute distance=0;strictly below epsilon=Yes

recorded pair2,3 signed difference := sub(1/6, 1/7)

recorded pair2,3 signed difference = 1/42

ordered recorded indices=[2, 3];signed difference=1/42;epsilon=1/5

absolute distance=1/42;strictly below epsilon=Yes

recorded pair2,4 signed difference := sub(1/6, 1/8)

recorded pair2,4 signed difference = 1/24

ordered recorded indices=[2, 4];signed difference=1/24;epsilon=1/5

absolute distance=1/24;strictly below epsilon=Yes

recorded pair3,2 signed difference := sub(1/7, 1/6)

recorded pair3,2 signed difference = -1/42

ordered recorded indices=[3, 2];signed difference=-1/42;epsilon=1/5

absolute distance=1/42;strictly below epsilon=Yes

recorded pair3,3 signed difference := sub(1/7, 1/7)

recorded pair3,3 signed difference = 0

ordered recorded indices=[3, 3];signed difference=0;epsilon=1/5

absolute distance=0;strictly below epsilon=Yes

recorded pair3,4 signed difference := sub(1/7, 1/8)

recorded pair3,4 signed difference = 1/56

ordered recorded indices=[3, 4];signed difference=1/56;epsilon=1/5

absolute distance=1/56;strictly below epsilon=Yes

recorded pair4,2 signed difference := sub(1/8, 1/6)

recorded pair4,2 signed difference = -1/24

ordered recorded indices=[4, 2];signed difference=-1/24;epsilon=1/5

absolute distance=1/24;strictly below epsilon=Yes

recorded pair4,3 signed difference := sub(1/8, 1/7)

recorded pair4,3 signed difference = -1/56

ordered recorded indices=[4, 3];signed difference=-1/56;epsilon=1/5

absolute distance=1/56;strictly below epsilon=Yes

recorded pair4,4 signed difference := sub(1/8, 1/8)

recorded pair4,4 signed difference = 0

ordered recorded indices=[4, 4];signed difference=0;epsilon=1/5

absolute distance=0;strictly below epsilon=Yes

all bad recorded pairs=[]

finite recorded tail passes=Yes;general Cauchy established=No

Reverse checks

• S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S018 · premise_and_result_audit — Recompute recorded pair4,4 signed difference with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S016 · premise_and_result_audit — Recompute recorded pair4,3 signed difference with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Recompute recorded pair4,2 signed difference with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Recompute recorded pair3,4 signed difference with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Recompute recorded pair3,3 signed difference with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Recompute recorded pair3,2 signed difference with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Recompute recorded pair2,4 signed difference with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Recompute recorded pair2,3 signed difference with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Recompute recorded pair2,2 signed difference with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([1/4, 1/5, 1/6, 1/7, 1/8], 1/5, 2, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)

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.