Question 12137: 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=[1/13, 1/14, 1/15, 1/16, 1/17], epsilon=1/4, 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=[1/13, 1/14, 1/15, 1/16, 1/17], epsilon=1/4, 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, 13], [1, 14], [1, 15], [1, 16], [1, 17]];raw epsilon=[1, 4]

normalized sequence=[1/13, 1/14, 1/15, 1/16, 1/17];positive epsilon=1/4;cutoff=2

recorded pair2,2 signed difference := sub(1/15, 1/15)

recorded pair2,2 signed difference = 0

ordered recorded indices=[2, 2];signed difference=0;epsilon=1/4

absolute distance=0;strictly below epsilon=Yes

recorded pair2,3 signed difference := sub(1/15, 1/16)

recorded pair2,3 signed difference = 1/240

ordered recorded indices=[2, 3];signed difference=1/240;epsilon=1/4

absolute distance=1/240;strictly below epsilon=Yes

recorded pair2,4 signed difference := sub(1/15, 1/17)

recorded pair2,4 signed difference = 2/255

ordered recorded indices=[2, 4];signed difference=2/255;epsilon=1/4

absolute distance=2/255;strictly below epsilon=Yes

recorded pair3,2 signed difference := sub(1/16, 1/15)

recorded pair3,2 signed difference = -1/240

ordered recorded indices=[3, 2];signed difference=-1/240;epsilon=1/4

absolute distance=1/240;strictly below epsilon=Yes

recorded pair3,3 signed difference := sub(1/16, 1/16)

recorded pair3,3 signed difference = 0

ordered recorded indices=[3, 3];signed difference=0;epsilon=1/4

absolute distance=0;strictly below epsilon=Yes

recorded pair3,4 signed difference := sub(1/16, 1/17)

recorded pair3,4 signed difference = 1/272

ordered recorded indices=[3, 4];signed difference=1/272;epsilon=1/4

absolute distance=1/272;strictly below epsilon=Yes

recorded pair4,2 signed difference := sub(1/17, 1/15)

recorded pair4,2 signed difference = -2/255

ordered recorded indices=[4, 2];signed difference=-2/255;epsilon=1/4

absolute distance=2/255;strictly below epsilon=Yes

recorded pair4,3 signed difference := sub(1/17, 1/16)

recorded pair4,3 signed difference = -1/272

ordered recorded indices=[4, 3];signed difference=-1/272;epsilon=1/4

absolute distance=1/272;strictly below epsilon=Yes

recorded pair4,4 signed difference := sub(1/17, 1/17)

recorded pair4,4 signed difference = 0

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

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

([1/13, 1/14, 1/15, 1/16, 1/17], 1/4, 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.