Question 12157: A Strict Reciprocal Bound for the Selected Epsilon

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=7, epsilon=1/9. 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=7, epsilon=1/9. 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

Step-by-step derivation

epsilon reciprocal threshold := div(9, 1)

epsilon reciprocal threshold = 9

positive epsilon=1/9;reciprocal threshold=9

floor threshold=9;9*1<=9<(9+1)*1

unclamped reciprocal cutoff := sum(9, -7, 1)

unclamped reciprocal cutoff = 3

reciprocal tail bound denominator := sum(3, 7)

reciprocal tail bound denominator = 10

reciprocal tail bound := div(1, 10)

reciprocal tail bound = 1/10

previous cutoff denominator := sum(2, 7)

previous cutoff denominator = 9

previous cutoff bound := div(1, 9)

previous cutoff bound = 1/9

N=3;bound=1/10;previous bound=1/9;epsilon=1/9

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

explicit x_n=1/(n+7);N=3;bound=1/10<epsilon=1/9

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

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

(7, 1/9, 3, 1/10, 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.