Question 12129: 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=[1/5, 1/6, 1/7, 1/8, 1/9], epsilon=1/6, cutoff N=0; 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=[1/5, 1/6, 1/7, 1/8, 1/9], epsilon=1/6, cutoff N=0; 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, 5], [1, 6], [1, 7], [1, 8], [1, 9]];raw epsilon=[1, 6]

normalized sequence=[1/5, 1/6, 1/7, 1/8, 1/9];positive epsilon=1/6;cutoff=0

recorded pair0,0 signed difference := sub(1/5, 1/5)

recorded pair0,0 signed difference = 0

ordered recorded indices=[0, 0];signed difference=0;epsilon=1/6

absolute distance=0;strictly below epsilon=Yes

recorded pair0,1 signed difference := sub(1/5, 1/6)

recorded pair0,1 signed difference = 1/30

ordered recorded indices=[0, 1];signed difference=1/30;epsilon=1/6

absolute distance=1/30;strictly below epsilon=Yes

recorded pair0,2 signed difference := sub(1/5, 1/7)

recorded pair0,2 signed difference = 2/35

ordered recorded indices=[0, 2];signed difference=2/35;epsilon=1/6

absolute distance=2/35;strictly below epsilon=Yes

recorded pair0,3 signed difference := sub(1/5, 1/8)

recorded pair0,3 signed difference = 3/40

ordered recorded indices=[0, 3];signed difference=3/40;epsilon=1/6

absolute distance=3/40;strictly below epsilon=Yes

recorded pair0,4 signed difference := sub(1/5, 1/9)

recorded pair0,4 signed difference = 4/45

ordered recorded indices=[0, 4];signed difference=4/45;epsilon=1/6

absolute distance=4/45;strictly below epsilon=Yes

recorded pair1,0 signed difference := sub(1/6, 1/5)

recorded pair1,0 signed difference = -1/30

ordered recorded indices=[1, 0];signed difference=-1/30;epsilon=1/6

absolute distance=1/30;strictly below epsilon=Yes

recorded pair1,1 signed difference := sub(1/6, 1/6)

recorded pair1,1 signed difference = 0

ordered recorded indices=[1, 1];signed difference=0;epsilon=1/6

absolute distance=0;strictly below epsilon=Yes

recorded pair1,2 signed difference := sub(1/6, 1/7)

recorded pair1,2 signed difference = 1/42

ordered recorded indices=[1, 2];signed difference=1/42;epsilon=1/6

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

recorded pair1,3 signed difference := sub(1/6, 1/8)

recorded pair1,3 signed difference = 1/24

ordered recorded indices=[1, 3];signed difference=1/24;epsilon=1/6

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

recorded pair1,4 signed difference := sub(1/6, 1/9)

recorded pair1,4 signed difference = 1/18

ordered recorded indices=[1, 4];signed difference=1/18;epsilon=1/6

absolute distance=1/18;strictly below epsilon=Yes

recorded pair2,0 signed difference := sub(1/7, 1/5)

recorded pair2,0 signed difference = -2/35

ordered recorded indices=[2, 0];signed difference=-2/35;epsilon=1/6

absolute distance=2/35;strictly below epsilon=Yes

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

recorded pair2,1 signed difference = -1/42

ordered recorded indices=[2, 1];signed difference=-1/42;epsilon=1/6

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

recorded pair2,2 signed difference := sub(1/7, 1/7)

recorded pair2,2 signed difference = 0

ordered recorded indices=[2, 2];signed difference=0;epsilon=1/6

absolute distance=0;strictly below epsilon=Yes

recorded pair2,3 signed difference := sub(1/7, 1/8)

recorded pair2,3 signed difference = 1/56

ordered recorded indices=[2, 3];signed difference=1/56;epsilon=1/6

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

recorded pair2,4 signed difference := sub(1/7, 1/9)

recorded pair2,4 signed difference = 2/63

ordered recorded indices=[2, 4];signed difference=2/63;epsilon=1/6

absolute distance=2/63;strictly below epsilon=Yes

recorded pair3,0 signed difference := sub(1/8, 1/5)

recorded pair3,0 signed difference = -3/40

ordered recorded indices=[3, 0];signed difference=-3/40;epsilon=1/6

absolute distance=3/40;strictly below epsilon=Yes

recorded pair3,1 signed difference := sub(1/8, 1/6)

recorded pair3,1 signed difference = -1/24

ordered recorded indices=[3, 1];signed difference=-1/24;epsilon=1/6

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

recorded pair3,2 signed difference := sub(1/8, 1/7)

recorded pair3,2 signed difference = -1/56

ordered recorded indices=[3, 2];signed difference=-1/56;epsilon=1/6

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

recorded pair3,3 signed difference := sub(1/8, 1/8)

recorded pair3,3 signed difference = 0

ordered recorded indices=[3, 3];signed difference=0;epsilon=1/6

absolute distance=0;strictly below epsilon=Yes

recorded pair3,4 signed difference := sub(1/8, 1/9)

recorded pair3,4 signed difference = 1/72

ordered recorded indices=[3, 4];signed difference=1/72;epsilon=1/6

absolute distance=1/72;strictly below epsilon=Yes

recorded pair4,0 signed difference := sub(1/9, 1/5)

recorded pair4,0 signed difference = -4/45

ordered recorded indices=[4, 0];signed difference=-4/45;epsilon=1/6

absolute distance=4/45;strictly below epsilon=Yes

recorded pair4,1 signed difference := sub(1/9, 1/6)

recorded pair4,1 signed difference = -1/18

ordered recorded indices=[4, 1];signed difference=-1/18;epsilon=1/6

absolute distance=1/18;strictly below epsilon=Yes

recorded pair4,2 signed difference := sub(1/9, 1/7)

recorded pair4,2 signed difference = -2/63

ordered recorded indices=[4, 2];signed difference=-2/63;epsilon=1/6

absolute distance=2/63;strictly below epsilon=Yes

recorded pair4,3 signed difference := sub(1/9, 1/8)

recorded pair4,3 signed difference = -1/72

ordered recorded indices=[4, 3];signed difference=-1/72;epsilon=1/6

absolute distance=1/72;strictly below epsilon=Yes

recorded pair4,4 signed difference := sub(1/9, 1/9)

recorded pair4,4 signed difference = 0

ordered recorded indices=[4, 4];signed difference=0;epsilon=1/6

absolute distance=0;strictly below epsilon=Yes

all bad recorded pairs=[]

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

Reverse checks

• S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S050 · premise_and_result_audit — Recompute recorded pair4,4 signed difference with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S048 · premise_and_result_audit — Recompute recorded pair4,3 signed difference with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S046 · premise_and_result_audit — Recompute recorded pair4,2 signed difference with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S044 · premise_and_result_audit — Recompute recorded pair4,1 signed difference with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S042 · premise_and_result_audit — Recompute recorded pair4,0 signed difference with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S040 · premise_and_result_audit — Recompute recorded pair3,4 signed difference with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S038 · premise_and_result_audit — Recompute recorded pair3,3 signed difference with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S036 · premise_and_result_audit — Recompute recorded pair3,2 signed difference with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S034 · premise_and_result_audit — Recompute recorded pair3,1 signed difference with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S032 · premise_and_result_audit — Recompute recorded pair3,0 signed difference with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S030 · premise_and_result_audit — Recompute recorded pair2,4 signed difference with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S028 · premise_and_result_audit — Recompute recorded pair2,3 signed difference with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S026 · premise_and_result_audit — Recompute recorded pair2,2 signed difference with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S024 · premise_and_result_audit — Recompute recorded pair2,1 signed difference with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S022 · premise_and_result_audit — Recompute recorded pair2,0 signed difference with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Recompute recorded pair1,4 signed difference with exact rational arithmetic, not rounded decimals.
• 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 pair1,3 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 pair1,2 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 pair1,1 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 pair1,0 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 pair0,4 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 pair0,3 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 pair0,2 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 pair0,1 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 pair0,0 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

([1/5, 1/6, 1/7, 1/8, 1/9], 1/6, 0, [], 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.