OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 12135: A Finite Tail Audit Cannot Establish General Cauchy Behavior

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

Return to catalogue page 61

← Question derivation index · Original question batch

R0023-12135 · q-20261013-advance-20261026-r03-cauchy-sequence-0010

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=[1/11, 1/12, 1/13, 1/14, 1/15], epsilon=1/7, 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=[1/11, 1/12, 1/13, 1/14, 1/15], epsilon=1/7, 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.

Step-by-step derivation

S001 · implies

Before

raw sequence=[[1, 11], [1, 12], [1, 13], [1, 14], [1, 15]];raw epsilon=[1, 7]

After

normalized sequence=[1/11, 1/12, 1/13, 1/14, 1/15];positive epsilon=1/7;cutoff=0

Conditions: none

Original denominators are nonzero and normalized epsilon must be positive.Only recorded indices are audited;N may equal the length,leaving an empty tail.

Step identifiers and dependencies

Rule: finite_Cauchy_exact_data_and_cutoff · Uses: G001

S002 · calculation

Before

recorded pair0,0 signed difference := sub(1/11, 1/11)

After

recorded pair0,0 signed difference = 0

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · implies

Before

ordered recorded indices=[0, 0];signed difference=0;epsilon=1/7

After

absolute distance=0;strictly below epsilon=Yes

Conditions: none

Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms.

Step identifiers and dependencies

Rule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S002

S004 · calculation

Before

recorded pair0,1 signed difference := sub(1/11, 1/12)

After

recorded pair0,1 signed difference = 1/132

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

ordered recorded indices=[0, 1];signed difference=1/132;epsilon=1/7

After

absolute distance=1/132;strictly below epsilon=Yes

Conditions: none

Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms.

Step identifiers and dependencies

Rule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S004

S006 · calculation

Before

recorded pair0,2 signed difference := sub(1/11, 1/13)

After

recorded pair0,2 signed difference = 2/143

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · implies

Before

ordered recorded indices=[0, 2];signed difference=2/143;epsilon=1/7

After

absolute distance=2/143;strictly below epsilon=Yes

Conditions: none

Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms.

Step identifiers and dependencies

Rule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S006

S008 · calculation

Before

recorded pair0,3 signed difference := sub(1/11, 1/14)

After

recorded pair0,3 signed difference = 3/154

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · implies

Before

ordered recorded indices=[0, 3];signed difference=3/154;epsilon=1/7

After

absolute distance=3/154;strictly below epsilon=Yes

Conditions: none

Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms.

Step identifiers and dependencies

Rule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S008

S010 · calculation

Before

recorded pair0,4 signed difference := sub(1/11, 1/15)

After

recorded pair0,4 signed difference = 4/165

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · implies

Before

ordered recorded indices=[0, 4];signed difference=4/165;epsilon=1/7

After

absolute distance=4/165;strictly below epsilon=Yes

Conditions: none

Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms.

Step identifiers and dependencies

Rule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S010

S012 · calculation

Before

recorded pair1,0 signed difference := sub(1/12, 1/11)

After

recorded pair1,0 signed difference = -1/132

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · implies

Before

ordered recorded indices=[1, 0];signed difference=-1/132;epsilon=1/7

After

absolute distance=1/132;strictly below epsilon=Yes

Conditions: none

Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms.

Step identifiers and dependencies

Rule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S012

S014 · calculation

Before

recorded pair1,1 signed difference := sub(1/12, 1/12)

After

recorded pair1,1 signed difference = 0

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · implies

Before

ordered recorded indices=[1, 1];signed difference=0;epsilon=1/7

After

absolute distance=0;strictly below epsilon=Yes

Conditions: none

Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms.

Step identifiers and dependencies

Rule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S014

S016 · calculation

Before

recorded pair1,2 signed difference := sub(1/12, 1/13)

After

recorded pair1,2 signed difference = 1/156

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · implies

Before

ordered recorded indices=[1, 2];signed difference=1/156;epsilon=1/7

After

absolute distance=1/156;strictly below epsilon=Yes

Conditions: none

Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms.

Step identifiers and dependencies

Rule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S016

S018 · calculation

Before

recorded pair1,3 signed difference := sub(1/12, 1/14)

After

recorded pair1,3 signed difference = 1/84

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · implies

Before

ordered recorded indices=[1, 3];signed difference=1/84;epsilon=1/7

After

absolute distance=1/84;strictly below epsilon=Yes

Conditions: none

Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms.

Step identifiers and dependencies

Rule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S018

S020 · calculation

Before

recorded pair1,4 signed difference := sub(1/12, 1/15)

After

recorded pair1,4 signed difference = 1/60

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · implies

Before

ordered recorded indices=[1, 4];signed difference=1/60;epsilon=1/7

After

absolute distance=1/60;strictly below epsilon=Yes

Conditions: none

Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms.

Step identifiers and dependencies

Rule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S020

S022 · calculation

Before

recorded pair2,0 signed difference := sub(1/13, 1/11)

After

recorded pair2,0 signed difference = -2/143

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · implies

Before

ordered recorded indices=[2, 0];signed difference=-2/143;epsilon=1/7

After

absolute distance=2/143;strictly below epsilon=Yes

Conditions: none

Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms.

Step identifiers and dependencies

Rule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S022

S024 · calculation

Before

recorded pair2,1 signed difference := sub(1/13, 1/12)

After

recorded pair2,1 signed difference = -1/156

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · implies

Before

ordered recorded indices=[2, 1];signed difference=-1/156;epsilon=1/7

After

absolute distance=1/156;strictly below epsilon=Yes

Conditions: none

Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms.

Step identifiers and dependencies

Rule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S024

S026 · calculation

Before

recorded pair2,2 signed difference := sub(1/13, 1/13)

After

recorded pair2,2 signed difference = 0

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · implies

Before

ordered recorded indices=[2, 2];signed difference=0;epsilon=1/7

After

absolute distance=0;strictly below epsilon=Yes

Conditions: none

Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms.

Step identifiers and dependencies

Rule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S026

S028 · calculation

Before

recorded pair2,3 signed difference := sub(1/13, 1/14)

After

recorded pair2,3 signed difference = 1/182

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · implies

Before

ordered recorded indices=[2, 3];signed difference=1/182;epsilon=1/7

After

absolute distance=1/182;strictly below epsilon=Yes

Conditions: none

Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms.

Step identifiers and dependencies

Rule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S028

S030 · calculation

Before

recorded pair2,4 signed difference := sub(1/13, 1/15)

After

recorded pair2,4 signed difference = 2/195

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · implies

Before

ordered recorded indices=[2, 4];signed difference=2/195;epsilon=1/7

After

absolute distance=2/195;strictly below epsilon=Yes

Conditions: none

Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms.

Step identifiers and dependencies

Rule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S030

S032 · calculation

Before

recorded pair3,0 signed difference := sub(1/14, 1/11)

After

recorded pair3,0 signed difference = -3/154

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · implies

Before

ordered recorded indices=[3, 0];signed difference=-3/154;epsilon=1/7

After

absolute distance=3/154;strictly below epsilon=Yes

Conditions: none

Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms.

Step identifiers and dependencies

Rule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S032

S034 · calculation

Before

recorded pair3,1 signed difference := sub(1/14, 1/12)

After

recorded pair3,1 signed difference = -1/84

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · implies

Before

ordered recorded indices=[3, 1];signed difference=-1/84;epsilon=1/7

After

absolute distance=1/84;strictly below epsilon=Yes

Conditions: none

Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms.

Step identifiers and dependencies

Rule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S034

S036 · calculation

Before

recorded pair3,2 signed difference := sub(1/14, 1/13)

After

recorded pair3,2 signed difference = -1/182

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · implies

Before

ordered recorded indices=[3, 2];signed difference=-1/182;epsilon=1/7

After

absolute distance=1/182;strictly below epsilon=Yes

Conditions: none

Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms.

Step identifiers and dependencies

Rule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S036

S038 · calculation

Before

recorded pair3,3 signed difference := sub(1/14, 1/14)

After

recorded pair3,3 signed difference = 0

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · implies

Before

ordered recorded indices=[3, 3];signed difference=0;epsilon=1/7

After

absolute distance=0;strictly below epsilon=Yes

Conditions: none

Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms.

Step identifiers and dependencies

Rule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S038

S040 · calculation

Before

recorded pair3,4 signed difference := sub(1/14, 1/15)

After

recorded pair3,4 signed difference = 1/210

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · implies

Before

ordered recorded indices=[3, 4];signed difference=1/210;epsilon=1/7

After

absolute distance=1/210;strictly below epsilon=Yes

Conditions: none

Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms.

Step identifiers and dependencies

Rule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S040

S042 · calculation

Before

recorded pair4,0 signed difference := sub(1/15, 1/11)

After

recorded pair4,0 signed difference = -4/165

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · implies

Before

ordered recorded indices=[4, 0];signed difference=-4/165;epsilon=1/7

After

absolute distance=4/165;strictly below epsilon=Yes

Conditions: none

Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms.

Step identifiers and dependencies

Rule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S042

S044 · calculation

Before

recorded pair4,1 signed difference := sub(1/15, 1/12)

After

recorded pair4,1 signed difference = -1/60

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · implies

Before

ordered recorded indices=[4, 1];signed difference=-1/60;epsilon=1/7

After

absolute distance=1/60;strictly below epsilon=Yes

Conditions: none

Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms.

Step identifiers and dependencies

Rule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S044

S046 · calculation

Before

recorded pair4,2 signed difference := sub(1/15, 1/13)

After

recorded pair4,2 signed difference = -2/195

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S045

S047 · implies

Before

ordered recorded indices=[4, 2];signed difference=-2/195;epsilon=1/7

After

absolute distance=2/195;strictly below epsilon=Yes

Conditions: none

Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms.

Step identifiers and dependencies

Rule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S046

S048 · calculation

Before

recorded pair4,3 signed difference := sub(1/15, 1/14)

After

recorded pair4,3 signed difference = -1/210

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S047

S049 · implies

Before

ordered recorded indices=[4, 3];signed difference=-1/210;epsilon=1/7

After

absolute distance=1/210;strictly below epsilon=Yes

Conditions: none

Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms.

Step identifiers and dependencies

Rule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S048

S050 · calculation

Before

recorded pair4,4 signed difference := sub(1/15, 1/15)

After

recorded pair4,4 signed difference = 0

Conditions: none

Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S049

S051 · implies

Before

ordered recorded indices=[4, 4];signed difference=0;epsilon=1/7

After

absolute distance=0;strictly below epsilon=Yes

Conditions: none

Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms.

Step identifiers and dependencies

Rule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S050

S052 · implies

Before

all bad recorded pairs=[]

After

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

Conditions: none

An empty tail can pass vacuously.Finite records do not supply the infinite for-every-epsilon/eventual-all-pairs obligation.No here means not established,not that every infinite continuation is non-Cauchy.

Step identifiers and dependencies

Rule: finite_tail_not_quantified_infinite_Cauchy · Uses: G001, S051

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

([1/11, 1/12, 1/13, 1/14, 1/15], 1/7, 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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: b3189feaf7a29154f6688408beaeaa32d864a9edb39d414561134cd18045e897
  • candidate record SHA-256: b0d68b72158cb9c9a6f20bccf97248653d328743809e7a85a910039bc1fa61f2
  • semantic review receipt SHA-256: f783e81f44ff0faa13dbf948657e9187244a59a809ebdf96f22cca24966649ff
  • question-first witness SHA-256: 49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON