# Question 12153: A Strict Reciprocal Bound for the Selected Epsilon

[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-0053)

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.

R0023-12153 · q-20261013-advance-20261026-r03-cauchy-sequence-0053



## 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. The infinite model is x_n=1/(n+shift), with shift=3, epsilon=1/5. Give (shift,epsilon,least nonnegative N with 1/(N+shift)<epsilon,bound,m,n>=N implies distance<=1/(N+shift)<epsilon,cauchy_for_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. The infinite model is x_n=1/(n+shift), with shift=3, epsilon=1/5. Give (shift,epsilon,least nonnegative N with 1/(N+shift)<epsilon,bound,m,n>=N implies distance<=1/(N+shift)<epsilon,cauchy_for_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 · calculationBefore

epsilon reciprocal threshold := div(5, 1)

After

epsilon reciprocal threshold = 5

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





S002 · impliesBefore

positive epsilon=1/5;reciprocal threshold=5

After

floor threshold=5;5*1<=5<(5+1)*1

Conditions: none

Since a and denominators are positive,1/(N+shift)<a/b is equivalent to N+shift>b/a.Use the exact floor to handle the strict boundary.

Step identifiers and dependenciesRule: exact_integer_threshold_floor · Uses: G001, S001





S003 · calculationBefore

unclamped reciprocal cutoff := sum(5, -3, 1)

After

unclamped reciprocal cutoff = 3

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, S002





S004 · calculationBefore

reciprocal tail bound denominator := sum(3, 3)

After

reciprocal tail bound denominator = 6

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 · calculationBefore

reciprocal tail bound := div(1, 6)

After

reciprocal tail bound = 1/6

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, S004





S006 · calculationBefore

previous cutoff denominator := sum(2, 3)

After

previous cutoff denominator = 5

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 · calculationBefore

previous cutoff bound := div(1, 5)

After

previous cutoff bound = 1/5

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, S006





S008 · impliesBefore

N=3;bound=1/6;previous bound=1/5;epsilon=1/5

After

current bound is strict;previous cutoff fails;N is least nonnegative

Conditions: none

Positive reciprocals decrease with the index.The previous bound is still≥epsilon,so every smaller nonnegative cutoff fails.Equality remains a failure.

Step identifiers and dependenciesRule: reciprocal_least_cutoff_strict_boundary · Uses: G001, S007





S009 · impliesBefore

explicit x_n=1/(n+3);N=3;bound=1/6<epsilon=1/5

After

all m,n>=N:both terms in[0,bound],so distance<=bound<epsilon

Conditions: none

Any two tail terms lie between0 and the bound,so their distance is at most that bound.The cutoff formula works for arbitrary positive epsilon;the output flag certifies this selected epsilon in the explicit model.

Step identifiers and dependenciesRule: reciprocal_all_tail_pairs_not_samples · Uses: G001, S008









## Reverse checks


- 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S007 · premise_and_result_audit — Recompute previous cutoff bound with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Recompute previous cutoff denominator with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Recompute reciprocal tail bound with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute reciprocal tail bound denominator with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute unclamped reciprocal cutoff with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Recompute epsilon reciprocal threshold with exact rational arithmetic, not rounded decimals.





## Result

(3, 1/5, 3, 1/6, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)

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: e1b9d388e064ff86187869a989d0863a7a5cbdb652eb388954c1dec5d254282a
- candidate record SHA-256: e4f7aeb244575f5248140f51ae3946065f70e95ca90b50ef0b6901282e3e6297
- 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-0053.json)