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

[Return to catalogue page 61](https://ocme.evemisslab.com/en/questions/page/61/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261026-r03-cauchy-sequence-foundations/#q-20261013-advance-20261026-r03-cauchy-sequence-0010)

This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

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.


- [mko-cauchy-sequence](https://ocme.evemisslab.com/en/mko/mko-cauchy-sequence/)





## Step-by-step derivation

S001 · impliesBefore

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 dependenciesRule: finite_Cauchy_exact_data_and_cutoff · Uses: G001





S002 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · impliesBefore

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 dependenciesRule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S002





S004 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · impliesBefore

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 dependenciesRule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S004





S006 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · impliesBefore

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 dependenciesRule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S006





S008 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · impliesBefore

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 dependenciesRule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S008





S010 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S009





S011 · impliesBefore

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 dependenciesRule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S010





S012 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S011





S013 · impliesBefore

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 dependenciesRule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S012





S014 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S013





S015 · impliesBefore

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 dependenciesRule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S014





S016 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S015





S017 · impliesBefore

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 dependenciesRule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S016





S018 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S017





S019 · impliesBefore

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 dependenciesRule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S018





S020 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S019





S021 · impliesBefore

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 dependenciesRule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S020





S022 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S021





S023 · impliesBefore

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 dependenciesRule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S022





S024 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S023





S025 · impliesBefore

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 dependenciesRule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S024





S026 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S025





S027 · impliesBefore

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 dependenciesRule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S026





S028 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S027





S029 · impliesBefore

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 dependenciesRule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S028





S030 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S029





S031 · impliesBefore

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 dependenciesRule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S030





S032 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S031





S033 · impliesBefore

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 dependenciesRule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S032





S034 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S033





S035 · impliesBefore

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 dependenciesRule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S034





S036 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S035





S037 · impliesBefore

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 dependenciesRule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S036





S038 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S037





S039 · impliesBefore

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 dependenciesRule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S038





S040 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S039





S041 · impliesBefore

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 dependenciesRule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S040





S042 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S041





S043 · impliesBefore

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 dependenciesRule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S042





S044 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S043





S045 · impliesBefore

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 dependenciesRule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S044





S046 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S045





S047 · impliesBefore

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 dependenciesRule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S046





S048 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S047





S049 · impliesBefore

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 dependenciesRule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S048





S050 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S049





S051 · impliesBefore

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 dependenciesRule: finite_Cauchy_each_ordered_tail_pair · Uses: G001, S050





S052 · impliesBefore

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 dependenciesRule: 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](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0010.json)