Question 12147: 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=[1/23, 1/24, 1/25, 1/26, 1/27], epsilon=1/4, 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=[1/23, 1/24, 1/25, 1/26, 1/27], epsilon=1/4, 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, 23], [1, 24], [1, 25], [1, 26], [1, 27]];raw epsilon=[1, 4]

normalized sequence=[1/23, 1/24, 1/25, 1/26, 1/27];positive epsilon=1/4;cutoff=0

recorded pair0,0 signed difference := sub(1/23, 1/23)

recorded pair0,0 signed difference = 0

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

absolute distance=0;strictly below epsilon=Yes

recorded pair0,1 signed difference := sub(1/23, 1/24)

recorded pair0,1 signed difference = 1/552

ordered recorded indices=[0, 1];signed difference=1/552;epsilon=1/4

absolute distance=1/552;strictly below epsilon=Yes

recorded pair0,2 signed difference := sub(1/23, 1/25)

recorded pair0,2 signed difference = 2/575

ordered recorded indices=[0, 2];signed difference=2/575;epsilon=1/4

absolute distance=2/575;strictly below epsilon=Yes

recorded pair0,3 signed difference := sub(1/23, 1/26)

recorded pair0,3 signed difference = 3/598

ordered recorded indices=[0, 3];signed difference=3/598;epsilon=1/4

absolute distance=3/598;strictly below epsilon=Yes

recorded pair0,4 signed difference := sub(1/23, 1/27)

recorded pair0,4 signed difference = 4/621

ordered recorded indices=[0, 4];signed difference=4/621;epsilon=1/4

absolute distance=4/621;strictly below epsilon=Yes

recorded pair1,0 signed difference := sub(1/24, 1/23)

recorded pair1,0 signed difference = -1/552

ordered recorded indices=[1, 0];signed difference=-1/552;epsilon=1/4

absolute distance=1/552;strictly below epsilon=Yes

recorded pair1,1 signed difference := sub(1/24, 1/24)

recorded pair1,1 signed difference = 0

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

absolute distance=0;strictly below epsilon=Yes

recorded pair1,2 signed difference := sub(1/24, 1/25)

recorded pair1,2 signed difference = 1/600

ordered recorded indices=[1, 2];signed difference=1/600;epsilon=1/4

absolute distance=1/600;strictly below epsilon=Yes

recorded pair1,3 signed difference := sub(1/24, 1/26)

recorded pair1,3 signed difference = 1/312

ordered recorded indices=[1, 3];signed difference=1/312;epsilon=1/4

absolute distance=1/312;strictly below epsilon=Yes

recorded pair1,4 signed difference := sub(1/24, 1/27)

recorded pair1,4 signed difference = 1/216

ordered recorded indices=[1, 4];signed difference=1/216;epsilon=1/4

absolute distance=1/216;strictly below epsilon=Yes

recorded pair2,0 signed difference := sub(1/25, 1/23)

recorded pair2,0 signed difference = -2/575

ordered recorded indices=[2, 0];signed difference=-2/575;epsilon=1/4

absolute distance=2/575;strictly below epsilon=Yes

recorded pair2,1 signed difference := sub(1/25, 1/24)

recorded pair2,1 signed difference = -1/600

ordered recorded indices=[2, 1];signed difference=-1/600;epsilon=1/4

absolute distance=1/600;strictly below epsilon=Yes

recorded pair2,2 signed difference := sub(1/25, 1/25)

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(1/25, 1/26)

recorded pair2,3 signed difference = 1/650

ordered recorded indices=[2, 3];signed difference=1/650;epsilon=1/4

absolute distance=1/650;strictly below epsilon=Yes

recorded pair2,4 signed difference := sub(1/25, 1/27)

recorded pair2,4 signed difference = 2/675

ordered recorded indices=[2, 4];signed difference=2/675;epsilon=1/4

absolute distance=2/675;strictly below epsilon=Yes

recorded pair3,0 signed difference := sub(1/26, 1/23)

recorded pair3,0 signed difference = -3/598

ordered recorded indices=[3, 0];signed difference=-3/598;epsilon=1/4

absolute distance=3/598;strictly below epsilon=Yes

recorded pair3,1 signed difference := sub(1/26, 1/24)

recorded pair3,1 signed difference = -1/312

ordered recorded indices=[3, 1];signed difference=-1/312;epsilon=1/4

absolute distance=1/312;strictly below epsilon=Yes

recorded pair3,2 signed difference := sub(1/26, 1/25)

recorded pair3,2 signed difference = -1/650

ordered recorded indices=[3, 2];signed difference=-1/650;epsilon=1/4

absolute distance=1/650;strictly below epsilon=Yes

recorded pair3,3 signed difference := sub(1/26, 1/26)

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(1/26, 1/27)

recorded pair3,4 signed difference = 1/702

ordered recorded indices=[3, 4];signed difference=1/702;epsilon=1/4

absolute distance=1/702;strictly below epsilon=Yes

recorded pair4,0 signed difference := sub(1/27, 1/23)

recorded pair4,0 signed difference = -4/621

ordered recorded indices=[4, 0];signed difference=-4/621;epsilon=1/4

absolute distance=4/621;strictly below epsilon=Yes

recorded pair4,1 signed difference := sub(1/27, 1/24)

recorded pair4,1 signed difference = -1/216

ordered recorded indices=[4, 1];signed difference=-1/216;epsilon=1/4

absolute distance=1/216;strictly below epsilon=Yes

recorded pair4,2 signed difference := sub(1/27, 1/25)

recorded pair4,2 signed difference = -2/675

ordered recorded indices=[4, 2];signed difference=-2/675;epsilon=1/4

absolute distance=2/675;strictly below epsilon=Yes

recorded pair4,3 signed difference := sub(1/27, 1/26)

recorded pair4,3 signed difference = -1/702

ordered recorded indices=[4, 3];signed difference=-1/702;epsilon=1/4

absolute distance=1/702;strictly below epsilon=Yes

recorded pair4,4 signed difference := sub(1/27, 1/27)

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

• 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

([1/23, 1/24, 1/25, 1/26, 1/27], 1/4, 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.