Finite Tail Audits, Infinite Cauchy Witnesses and Alternating Counterexamples
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261013-advance-20261026-r03-cauchy-sequence-foundations · 2026-10-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. A Finite Tail Audit Cannot Establish General Cauchy Behavior
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=[1/2, 1/3, 1/4, 1/5, 1/6], epsilon=1/3, 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.
View answer and explanation
([1/2, 1/3, 1/4, 1/5, 1/6], 1/3, 0, [[0, 4, 1/3], [4, 0, 1/3]], No, No, finite-tail-insufficient-for-general-Cauchy)
Enumerate every ordered pair of listed indices at or beyond N. An empty tail passes vacuously. The general_cauchy No means not established by this finite audit, not a proof that every infinite continuation is non-Cauchy. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1/2, 1/3, 1/4, 1/5, 1/6], 1/3, 0, [[0, 4, 1/3], [4, 0, 1/3]], No, No, finite-tail-insufficient-for-general-Cauchy).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-finite-tail-audit · candidate
2. A Finite Tail Audit Cannot Establish General Cauchy Behavior
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=[1/3, 1/4, 1/5, 1/6, 1/7], epsilon=1/4, cutoff N=1; 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.
View answer and explanation
([1/3, 1/4, 1/5, 1/6, 1/7], 1/4, 1, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)
Enumerate every ordered pair of listed indices at or beyond N. An empty tail passes vacuously. The general_cauchy No means not established by this finite audit, not a proof that every infinite continuation is non-Cauchy. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1/3, 1/4, 1/5, 1/6, 1/7], 1/4, 1, [], Yes, No, finite-tail-insufficient-for-general-Cauchy).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-finite-tail-audit · candidate
3. A Finite Tail Audit Cannot Establish General Cauchy Behavior
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=[1/4, 1/5, 1/6, 1/7, 1/8], epsilon=1/5, cutoff N=2; 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.
View answer and explanation
([1/4, 1/5, 1/6, 1/7, 1/8], 1/5, 2, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)
Enumerate every ordered pair of listed indices at or beyond N. An empty tail passes vacuously. The general_cauchy No means not established by this finite audit, not a proof that every infinite continuation is non-Cauchy. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1/4, 1/5, 1/6, 1/7, 1/8], 1/5, 2, [], Yes, No, finite-tail-insufficient-for-general-Cauchy).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-finite-tail-audit · candidate
4. A Finite Tail Audit Cannot Establish General Cauchy Behavior
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=[1/5, 1/6, 1/7, 1/8, 1/9], epsilon=1/6, 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.
View answer and explanation
([1/5, 1/6, 1/7, 1/8, 1/9], 1/6, 0, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)
Enumerate every ordered pair of listed indices at or beyond N. An empty tail passes vacuously. The general_cauchy No means not established by this finite audit, not a proof that every infinite continuation is non-Cauchy. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1/5, 1/6, 1/7, 1/8, 1/9], 1/6, 0, [], Yes, No, finite-tail-insufficient-for-general-Cauchy).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-finite-tail-audit · candidate
5. A Finite Tail Audit Cannot Establish General Cauchy Behavior
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=[1/6, 1/7, 1/8, 1/9, 1/10], epsilon=1/7, cutoff N=1; 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.
View answer and explanation
([1/6, 1/7, 1/8, 1/9, 1/10], 1/7, 1, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)
Enumerate every ordered pair of listed indices at or beyond N. An empty tail passes vacuously. The general_cauchy No means not established by this finite audit, not a proof that every infinite continuation is non-Cauchy. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1/6, 1/7, 1/8, 1/9, 1/10], 1/7, 1, [], Yes, No, finite-tail-insufficient-for-general-Cauchy).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-finite-tail-audit · candidate
6. A Finite Tail Audit Cannot Establish General Cauchy Behavior
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=[1/7, 1/8, 1/9, 1/10, 1/11], epsilon=1/3, cutoff N=2; 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.
View answer and explanation
([1/7, 1/8, 1/9, 1/10, 1/11], 1/3, 2, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)
Enumerate every ordered pair of listed indices at or beyond N. An empty tail passes vacuously. The general_cauchy No means not established by this finite audit, not a proof that every infinite continuation is non-Cauchy. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1/7, 1/8, 1/9, 1/10, 1/11], 1/3, 2, [], Yes, No, finite-tail-insufficient-for-general-Cauchy).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-finite-tail-audit · candidate
7. A Finite Tail Audit Cannot Establish General Cauchy Behavior
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=[1/8, 1/9, 1/10, 1/11, 1/12], 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.
View answer and explanation
([1/8, 1/9, 1/10, 1/11, 1/12], 1/4, 0, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)
Enumerate every ordered pair of listed indices at or beyond N. An empty tail passes vacuously. The general_cauchy No means not established by this finite audit, not a proof that every infinite continuation is non-Cauchy. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1/8, 1/9, 1/10, 1/11, 1/12], 1/4, 0, [], Yes, No, finite-tail-insufficient-for-general-Cauchy).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-finite-tail-audit · candidate
8. A Finite Tail Audit Cannot Establish General Cauchy Behavior
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=[1/9, 1/10, 1/11, 1/12, 1/13], epsilon=1/5, cutoff N=1; 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.
View answer and explanation
([1/9, 1/10, 1/11, 1/12, 1/13], 1/5, 1, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)
Enumerate every ordered pair of listed indices at or beyond N. An empty tail passes vacuously. The general_cauchy No means not established by this finite audit, not a proof that every infinite continuation is non-Cauchy. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1/9, 1/10, 1/11, 1/12, 1/13], 1/5, 1, [], Yes, No, finite-tail-insufficient-for-general-Cauchy).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-finite-tail-audit · candidate
9. A Finite Tail Audit Cannot Establish General Cauchy Behavior
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=[1/10, 1/11, 1/12, 1/13, 1/14], epsilon=1/6, cutoff N=2; 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.
View answer and explanation
([1/10, 1/11, 1/12, 1/13, 1/14], 1/6, 2, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)
Enumerate every ordered pair of listed indices at or beyond N. An empty tail passes vacuously. The general_cauchy No means not established by this finite audit, not a proof that every infinite continuation is non-Cauchy. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1/10, 1/11, 1/12, 1/13, 1/14], 1/6, 2, [], Yes, No, finite-tail-insufficient-for-general-Cauchy).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-finite-tail-audit · candidate
10. A Finite Tail Audit Cannot Establish General Cauchy Behavior
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.
View answer and explanation
([1/11, 1/12, 1/13, 1/14, 1/15], 1/7, 0, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)
Enumerate every ordered pair of listed indices at or beyond N. An empty tail passes vacuously. The general_cauchy No means not established by this finite audit, not a proof that every infinite continuation is non-Cauchy. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1/11, 1/12, 1/13, 1/14, 1/15], 1/7, 0, [], Yes, No, finite-tail-insufficient-for-general-Cauchy).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-finite-tail-audit · candidate
11. A Finite Tail Audit Cannot Establish General Cauchy Behavior
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=[1/12, 1/13, 1/14, 1/15, 1/16], epsilon=1/3, cutoff N=1; 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.
View answer and explanation
([1/12, 1/13, 1/14, 1/15, 1/16], 1/3, 1, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)
Enumerate every ordered pair of listed indices at or beyond N. An empty tail passes vacuously. The general_cauchy No means not established by this finite audit, not a proof that every infinite continuation is non-Cauchy. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1/12, 1/13, 1/14, 1/15, 1/16], 1/3, 1, [], Yes, No, finite-tail-insufficient-for-general-Cauchy).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-finite-tail-audit · candidate
12. A Finite Tail Audit Cannot Establish General Cauchy Behavior
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=[1/13, 1/14, 1/15, 1/16, 1/17], epsilon=1/4, cutoff N=2; 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.
View answer and explanation
([1/13, 1/14, 1/15, 1/16, 1/17], 1/4, 2, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)
Enumerate every ordered pair of listed indices at or beyond N. An empty tail passes vacuously. The general_cauchy No means not established by this finite audit, not a proof that every infinite continuation is non-Cauchy. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1/13, 1/14, 1/15, 1/16, 1/17], 1/4, 2, [], Yes, No, finite-tail-insufficient-for-general-Cauchy).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-finite-tail-audit · candidate
13. A Finite Tail Audit Cannot Establish General Cauchy Behavior
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=[1/14, 1/15, 1/16, 1/17, 1/18], epsilon=1/5, 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.
View answer and explanation
([1/14, 1/15, 1/16, 1/17, 1/18], 1/5, 0, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)
Enumerate every ordered pair of listed indices at or beyond N. An empty tail passes vacuously. The general_cauchy No means not established by this finite audit, not a proof that every infinite continuation is non-Cauchy. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1/14, 1/15, 1/16, 1/17, 1/18], 1/5, 0, [], Yes, No, finite-tail-insufficient-for-general-Cauchy).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-finite-tail-audit · candidate
14. A Finite Tail Audit Cannot Establish General Cauchy Behavior
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=[1/15, 1/16, 1/17, 1/18, 1/19], epsilon=1/6, cutoff N=1; 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.
View answer and explanation
([1/15, 1/16, 1/17, 1/18, 1/19], 1/6, 1, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)
Enumerate every ordered pair of listed indices at or beyond N. An empty tail passes vacuously. The general_cauchy No means not established by this finite audit, not a proof that every infinite continuation is non-Cauchy. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1/15, 1/16, 1/17, 1/18, 1/19], 1/6, 1, [], Yes, No, finite-tail-insufficient-for-general-Cauchy).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-finite-tail-audit · candidate
15. A Finite Tail Audit Cannot Establish General Cauchy Behavior
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=[1/16, 1/17, 1/18, 1/19, 1/20], epsilon=1/7, cutoff N=2; 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.
View answer and explanation
([1/16, 1/17, 1/18, 1/19, 1/20], 1/7, 2, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)
Enumerate every ordered pair of listed indices at or beyond N. An empty tail passes vacuously. The general_cauchy No means not established by this finite audit, not a proof that every infinite continuation is non-Cauchy. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1/16, 1/17, 1/18, 1/19, 1/20], 1/7, 2, [], Yes, No, finite-tail-insufficient-for-general-Cauchy).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-finite-tail-audit · candidate
16. A Finite Tail Audit Cannot Establish General Cauchy Behavior
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=[1/17, 1/18, 1/19, 1/20, 1/21], epsilon=1/3, 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.
View answer and explanation
([1/17, 1/18, 1/19, 1/20, 1/21], 1/3, 0, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)
Enumerate every ordered pair of listed indices at or beyond N. An empty tail passes vacuously. The general_cauchy No means not established by this finite audit, not a proof that every infinite continuation is non-Cauchy. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1/17, 1/18, 1/19, 1/20, 1/21], 1/3, 0, [], Yes, No, finite-tail-insufficient-for-general-Cauchy).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-finite-tail-audit · candidate
17. A Finite Tail Audit Cannot Establish General Cauchy Behavior
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=[1/18, 1/19, 1/20, 1/21, 1/22], epsilon=1/4, cutoff N=1; 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.
View answer and explanation
([1/18, 1/19, 1/20, 1/21, 1/22], 1/4, 1, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)
Enumerate every ordered pair of listed indices at or beyond N. An empty tail passes vacuously. The general_cauchy No means not established by this finite audit, not a proof that every infinite continuation is non-Cauchy. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1/18, 1/19, 1/20, 1/21, 1/22], 1/4, 1, [], Yes, No, finite-tail-insufficient-for-general-Cauchy).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-finite-tail-audit · candidate
18. A Finite Tail Audit Cannot Establish General Cauchy Behavior
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=[1/19, 1/20, 1/21, 1/22, 1/23], epsilon=1/5, cutoff N=2; 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.
View answer and explanation
([1/19, 1/20, 1/21, 1/22, 1/23], 1/5, 2, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)
Enumerate every ordered pair of listed indices at or beyond N. An empty tail passes vacuously. The general_cauchy No means not established by this finite audit, not a proof that every infinite continuation is non-Cauchy. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1/19, 1/20, 1/21, 1/22, 1/23], 1/5, 2, [], Yes, No, finite-tail-insufficient-for-general-Cauchy).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-finite-tail-audit · candidate
19. A Finite Tail Audit Cannot Establish General Cauchy Behavior
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=[1/20, 1/21, 1/22, 1/23, 1/24], epsilon=1/6, 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.
View answer and explanation
([1/20, 1/21, 1/22, 1/23, 1/24], 1/6, 0, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)
Enumerate every ordered pair of listed indices at or beyond N. An empty tail passes vacuously. The general_cauchy No means not established by this finite audit, not a proof that every infinite continuation is non-Cauchy. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1/20, 1/21, 1/22, 1/23, 1/24], 1/6, 0, [], Yes, No, finite-tail-insufficient-for-general-Cauchy).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-finite-tail-audit · candidate
20. A Finite Tail Audit Cannot Establish General Cauchy Behavior
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=[1/21, 1/22, 1/23, 1/24, 1/25], epsilon=1/7, cutoff N=1; 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.
View answer and explanation
([1/21, 1/22, 1/23, 1/24, 1/25], 1/7, 1, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)
Enumerate every ordered pair of listed indices at or beyond N. An empty tail passes vacuously. The general_cauchy No means not established by this finite audit, not a proof that every infinite continuation is non-Cauchy. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1/21, 1/22, 1/23, 1/24, 1/25], 1/7, 1, [], Yes, No, finite-tail-insufficient-for-general-Cauchy).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-finite-tail-audit · candidate
21. A Finite Tail Audit Cannot Establish General Cauchy Behavior
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=[1/22, 1/23, 1/24, 1/25, 1/26], epsilon=1/3, cutoff N=2; 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.
View answer and explanation
([1/22, 1/23, 1/24, 1/25, 1/26], 1/3, 2, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)
Enumerate every ordered pair of listed indices at or beyond N. An empty tail passes vacuously. The general_cauchy No means not established by this finite audit, not a proof that every infinite continuation is non-Cauchy. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1/22, 1/23, 1/24, 1/25, 1/26], 1/3, 2, [], Yes, No, finite-tail-insufficient-for-general-Cauchy).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-finite-tail-audit · candidate
22. A Finite Tail Audit Cannot Establish General Cauchy Behavior
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.
View answer and explanation
([1/23, 1/24, 1/25, 1/26, 1/27], 1/4, 0, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)
Enumerate every ordered pair of listed indices at or beyond N. An empty tail passes vacuously. The general_cauchy No means not established by this finite audit, not a proof that every infinite continuation is non-Cauchy. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1/23, 1/24, 1/25, 1/26, 1/27], 1/4, 0, [], Yes, No, finite-tail-insufficient-for-general-Cauchy).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-finite-tail-audit · candidate
23. A Finite Tail Audit Cannot Establish General Cauchy Behavior
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=[1/24, 1/25, 1/26, 1/27, 1/28], epsilon=1/5, cutoff N=1; 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.
View answer and explanation
([1/24, 1/25, 1/26, 1/27, 1/28], 1/5, 1, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)
Enumerate every ordered pair of listed indices at or beyond N. An empty tail passes vacuously. The general_cauchy No means not established by this finite audit, not a proof that every infinite continuation is non-Cauchy. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1/24, 1/25, 1/26, 1/27, 1/28], 1/5, 1, [], Yes, No, finite-tail-insufficient-for-general-Cauchy).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-finite-tail-audit · candidate
24. A Finite Tail Audit Cannot Establish General Cauchy Behavior
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=[1/25, 1/26, 1/27, 1/28, 1/29], epsilon=1/6, cutoff N=2; 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.
View answer and explanation
([1/25, 1/26, 1/27, 1/28, 1/29], 1/6, 2, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)
Enumerate every ordered pair of listed indices at or beyond N. An empty tail passes vacuously. The general_cauchy No means not established by this finite audit, not a proof that every infinite continuation is non-Cauchy. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1/25, 1/26, 1/27, 1/28, 1/29], 1/6, 2, [], Yes, No, finite-tail-insufficient-for-general-Cauchy).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-finite-tail-audit · candidate
25. A Finite Tail Audit Cannot Establish General Cauchy Behavior
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=[1/26, 1/27, 1/28, 1/29, 1/30], 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.
View answer and explanation
([1/26, 1/27, 1/28, 1/29, 1/30], 1/7, 0, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)
Enumerate every ordered pair of listed indices at or beyond N. An empty tail passes vacuously. The general_cauchy No means not established by this finite audit, not a proof that every infinite continuation is non-Cauchy. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1/26, 1/27, 1/28, 1/29, 1/30], 1/7, 0, [], Yes, No, finite-tail-insufficient-for-general-Cauchy).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-finite-tail-audit · candidate
26. An Explicit Infinite Constant Tail Supplies a Cauchy Witness
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 real sequence has finite prefix=[-12/1, 1/2, -3/1] followed forever by tail=-5/1. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.
View answer and explanation
([-12, 1/2, -3], -5, 3, for any epsilon>0 choose N=prefix length, Yes)
For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-12, 1/2, -3], -5, 3, for any epsilon>0 choose N=prefix length, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate
27. An Explicit Infinite Constant Tail Supplies a Cauchy Witness
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 real sequence has finite prefix=[-11/1, 2/2, -2/2] followed forever by tail=-4/2. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.
View answer and explanation
([-11, 1, -1], -2, 3, for any epsilon>0 choose N=prefix length, Yes)
For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-11, 1, -1], -2, 3, for any epsilon>0 choose N=prefix length, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate
28. An Explicit Infinite Constant Tail Supplies a Cauchy Witness
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 real sequence has finite prefix=[-10/1, 3/2, -1/3] followed forever by tail=-3/3. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.
View answer and explanation
([-10, 3/2, -1/3], -1, 3, for any epsilon>0 choose N=prefix length, Yes)
For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-10, 3/2, -1/3], -1, 3, for any epsilon>0 choose N=prefix length, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate
29. An Explicit Infinite Constant Tail Supplies a Cauchy Witness
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 real sequence has finite prefix=[-9/1, 4/2, 0/4] followed forever by tail=-2/4. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.
View answer and explanation
([-9, 2, 0], -1/2, 3, for any epsilon>0 choose N=prefix length, Yes)
For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-9, 2, 0], -1/2, 3, for any epsilon>0 choose N=prefix length, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate
30. An Explicit Infinite Constant Tail Supplies a Cauchy Witness
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 real sequence has finite prefix=[-8/1, 5/2, 1/5] followed forever by tail=-1/5. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.
View answer and explanation
([-8, 5/2, 1/5], -1/5, 3, for any epsilon>0 choose N=prefix length, Yes)
For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-8, 5/2, 1/5], -1/5, 3, for any epsilon>0 choose N=prefix length, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate
31. An Explicit Infinite Constant Tail Supplies a Cauchy Witness
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 real sequence has finite prefix=[-7/1, 6/2, 2/1] followed forever by tail=0/6. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.
View answer and explanation
([-7, 3, 2], 0, 3, for any epsilon>0 choose N=prefix length, Yes)
For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-7, 3, 2], 0, 3, for any epsilon>0 choose N=prefix length, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate
32. An Explicit Infinite Constant Tail Supplies a Cauchy Witness
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 real sequence has finite prefix=[-6/1, 7/2, 3/2] followed forever by tail=1/7. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.
View answer and explanation
([-6, 7/2, 3/2], 1/7, 3, for any epsilon>0 choose N=prefix length, Yes)
For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-6, 7/2, 3/2], 1/7, 3, for any epsilon>0 choose N=prefix length, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate
33. An Explicit Infinite Constant Tail Supplies a Cauchy Witness
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 real sequence has finite prefix=[-5/1, 8/2, -3/3] followed forever by tail=2/1. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.
View answer and explanation
([-5, 4, -1], 2, 3, for any epsilon>0 choose N=prefix length, Yes)
For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-5, 4, -1], 2, 3, for any epsilon>0 choose N=prefix length, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate
34. An Explicit Infinite Constant Tail Supplies a Cauchy Witness
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 real sequence has finite prefix=[-4/1, 9/2, -2/4] followed forever by tail=3/2. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.
View answer and explanation
([-4, 9/2, -1/2], 3/2, 3, for any epsilon>0 choose N=prefix length, Yes)
For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-4, 9/2, -1/2], 3/2, 3, for any epsilon>0 choose N=prefix length, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate
35. An Explicit Infinite Constant Tail Supplies a Cauchy Witness
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 real sequence has finite prefix=[-3/1, 10/2, -1/5] followed forever by tail=4/3. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.
View answer and explanation
([-3, 5, -1/5], 4/3, 3, for any epsilon>0 choose N=prefix length, Yes)
For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-3, 5, -1/5], 4/3, 3, for any epsilon>0 choose N=prefix length, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate
36. An Explicit Infinite Constant Tail Supplies a Cauchy Witness
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 real sequence has finite prefix=[-2/1, 11/2, 0/1] followed forever by tail=5/4. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.
View answer and explanation
([-2, 11/2, 0], 5/4, 3, for any epsilon>0 choose N=prefix length, Yes)
For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-2, 11/2, 0], 5/4, 3, for any epsilon>0 choose N=prefix length, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate
37. An Explicit Infinite Constant Tail Supplies a Cauchy Witness
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 real sequence has finite prefix=[-1/1, 12/2, 1/2] followed forever by tail=-5/5. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.
View answer and explanation
([-1, 6, 1/2], -1, 3, for any epsilon>0 choose N=prefix length, Yes)
For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-1, 6, 1/2], -1, 3, for any epsilon>0 choose N=prefix length, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate
38. An Explicit Infinite Constant Tail Supplies a Cauchy Witness
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 real sequence has finite prefix=[0/1, 13/2, 2/3] followed forever by tail=-4/6. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.
View answer and explanation
([0, 13/2, 2/3], -2/3, 3, for any epsilon>0 choose N=prefix length, Yes)
For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 13/2, 2/3], -2/3, 3, for any epsilon>0 choose N=prefix length, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate
39. An Explicit Infinite Constant Tail Supplies a Cauchy Witness
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 real sequence has finite prefix=[1/1, 14/2, 3/4] followed forever by tail=-3/7. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.
View answer and explanation
([1, 7, 3/4], -3/7, 3, for any epsilon>0 choose N=prefix length, Yes)
For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 7, 3/4], -3/7, 3, for any epsilon>0 choose N=prefix length, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate
40. An Explicit Infinite Constant Tail Supplies a Cauchy Witness
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 real sequence has finite prefix=[2/1, 15/2, -3/5] followed forever by tail=-2/1. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.
View answer and explanation
([2, 15/2, -3/5], -2, 3, for any epsilon>0 choose N=prefix length, Yes)
For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 15/2, -3/5], -2, 3, for any epsilon>0 choose N=prefix length, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate
41. An Explicit Infinite Constant Tail Supplies a Cauchy Witness
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 real sequence has finite prefix=[3/1, 16/2, -2/1] followed forever by tail=-1/2. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.
View answer and explanation
([3, 8, -2], -1/2, 3, for any epsilon>0 choose N=prefix length, Yes)
For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([3, 8, -2], -1/2, 3, for any epsilon>0 choose N=prefix length, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate
42. An Explicit Infinite Constant Tail Supplies a Cauchy Witness
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 real sequence has finite prefix=[4/1, 17/2, -1/2] followed forever by tail=0/3. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.
View answer and explanation
([4, 17/2, -1/2], 0, 3, for any epsilon>0 choose N=prefix length, Yes)
For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([4, 17/2, -1/2], 0, 3, for any epsilon>0 choose N=prefix length, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate
43. An Explicit Infinite Constant Tail Supplies a Cauchy Witness
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 real sequence has finite prefix=[5/1, 18/2, 0/3] followed forever by tail=1/4. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.
View answer and explanation
([5, 9, 0], 1/4, 3, for any epsilon>0 choose N=prefix length, Yes)
For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([5, 9, 0], 1/4, 3, for any epsilon>0 choose N=prefix length, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate
44. An Explicit Infinite Constant Tail Supplies a Cauchy Witness
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 real sequence has finite prefix=[6/1, 19/2, 1/4] followed forever by tail=2/5. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.
View answer and explanation
([6, 19/2, 1/4], 2/5, 3, for any epsilon>0 choose N=prefix length, Yes)
For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([6, 19/2, 1/4], 2/5, 3, for any epsilon>0 choose N=prefix length, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate
45. An Explicit Infinite Constant Tail Supplies a Cauchy Witness
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 real sequence has finite prefix=[7/1, 20/2, 2/5] followed forever by tail=3/6. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.
View answer and explanation
([7, 10, 2/5], 1/2, 3, for any epsilon>0 choose N=prefix length, Yes)
For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([7, 10, 2/5], 1/2, 3, for any epsilon>0 choose N=prefix length, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate
46. An Explicit Infinite Constant Tail Supplies a Cauchy Witness
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 real sequence has finite prefix=[8/1, 21/2, 3/1] followed forever by tail=4/7. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.
View answer and explanation
([8, 21/2, 3], 4/7, 3, for any epsilon>0 choose N=prefix length, Yes)
For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([8, 21/2, 3], 4/7, 3, for any epsilon>0 choose N=prefix length, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate
47. An Explicit Infinite Constant Tail Supplies a Cauchy Witness
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 real sequence has finite prefix=[9/1, 22/2, -3/2] followed forever by tail=5/1. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.
View answer and explanation
([9, 11, -3/2], 5, 3, for any epsilon>0 choose N=prefix length, Yes)
For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([9, 11, -3/2], 5, 3, for any epsilon>0 choose N=prefix length, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate
48. An Explicit Infinite Constant Tail Supplies a Cauchy Witness
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 real sequence has finite prefix=[10/1, 23/2, -2/3] followed forever by tail=-5/2. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.
View answer and explanation
([10, 23/2, -2/3], -5/2, 3, for any epsilon>0 choose N=prefix length, Yes)
For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([10, 23/2, -2/3], -5/2, 3, for any epsilon>0 choose N=prefix length, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate
49. An Explicit Infinite Constant Tail Supplies a Cauchy Witness
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 real sequence has finite prefix=[11/1, 24/2, -1/4] followed forever by tail=-4/3. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.
View answer and explanation
([11, 12, -1/4], -4/3, 3, for any epsilon>0 choose N=prefix length, Yes)
For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([11, 12, -1/4], -4/3, 3, for any epsilon>0 choose N=prefix length, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate
50. An Explicit Infinite Constant Tail Supplies a Cauchy Witness
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 real sequence has finite prefix=[12/1, 25/2, 0/5] followed forever by tail=-3/4. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.
View answer and explanation
([12, 25/2, 0], -3/4, 3, for any epsilon>0 choose N=prefix length, Yes)
For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([12, 25/2, 0], -3/4, 3, for any epsilon>0 choose N=prefix length, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate
51. A Strict Reciprocal Bound for the Selected Epsilon
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=1, epsilon=1/3. 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.
View answer and explanation
(1, 1/3, 3, 1/4, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)
Retain the strict inequality when choosing N. For every two later indices the reciprocal terms lie between zero and the stated upper bound, so their distance is at most that bound. The output certifies this chosen epsilon and the explicit model, not an arbitrary sequence. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 1/3, 3, 1/4, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-reciprocal-epsilon-witness · candidate
52. A Strict Reciprocal Bound for the Selected Epsilon
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=2, epsilon=1/4. 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.
View answer and explanation
(2, 1/4, 3, 1/5, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)
Retain the strict inequality when choosing N. For every two later indices the reciprocal terms lie between zero and the stated upper bound, so their distance is at most that bound. The output certifies this chosen epsilon and the explicit model, not an arbitrary sequence. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 1/4, 3, 1/5, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-reciprocal-epsilon-witness · candidate
53. A Strict Reciprocal Bound for the Selected Epsilon
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.
View answer and explanation
(3, 1/5, 3, 1/6, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)
Retain the strict inequality when choosing N. For every two later indices the reciprocal terms lie between zero and the stated upper bound, so their distance is at most that bound. The output certifies this chosen epsilon and the explicit model, not an arbitrary sequence. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 1/5, 3, 1/6, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-reciprocal-epsilon-witness · candidate
54. A Strict Reciprocal Bound for the Selected Epsilon
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=4, epsilon=1/6. 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.
View answer and explanation
(4, 1/6, 3, 1/7, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)
Retain the strict inequality when choosing N. For every two later indices the reciprocal terms lie between zero and the stated upper bound, so their distance is at most that bound. The output certifies this chosen epsilon and the explicit model, not an arbitrary sequence. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 1/6, 3, 1/7, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-reciprocal-epsilon-witness · candidate
55. A Strict Reciprocal Bound for the Selected Epsilon
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=5, epsilon=1/7. 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.
View answer and explanation
(5, 1/7, 3, 1/8, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)
Retain the strict inequality when choosing N. For every two later indices the reciprocal terms lie between zero and the stated upper bound, so their distance is at most that bound. The output certifies this chosen epsilon and the explicit model, not an arbitrary sequence. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 1/7, 3, 1/8, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-reciprocal-epsilon-witness · candidate
56. A Strict Reciprocal Bound for the Selected Epsilon
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=6, epsilon=1/8. 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.
View answer and explanation
(6, 1/8, 3, 1/9, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)
Retain the strict inequality when choosing N. For every two later indices the reciprocal terms lie between zero and the stated upper bound, so their distance is at most that bound. The output certifies this chosen epsilon and the explicit model, not an arbitrary sequence. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 1/8, 3, 1/9, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-reciprocal-epsilon-witness · candidate
57. A Strict Reciprocal Bound for the Selected Epsilon
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.
View answer and explanation
(7, 1/9, 3, 1/10, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)
Retain the strict inequality when choosing N. For every two later indices the reciprocal terms lie between zero and the stated upper bound, so their distance is at most that bound. The output certifies this chosen epsilon and the explicit model, not an arbitrary sequence. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (7, 1/9, 3, 1/10, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-reciprocal-epsilon-witness · candidate
58. A Strict Reciprocal Bound for the Selected Epsilon
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=8, epsilon=1/10. 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.
View answer and explanation
(8, 1/10, 3, 1/11, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)
Retain the strict inequality when choosing N. For every two later indices the reciprocal terms lie between zero and the stated upper bound, so their distance is at most that bound. The output certifies this chosen epsilon and the explicit model, not an arbitrary sequence. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (8, 1/10, 3, 1/11, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-reciprocal-epsilon-witness · candidate
59. A Strict Reciprocal Bound for the Selected Epsilon
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=9, epsilon=1/11. 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.
View answer and explanation
(9, 1/11, 3, 1/12, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)
Retain the strict inequality when choosing N. For every two later indices the reciprocal terms lie between zero and the stated upper bound, so their distance is at most that bound. The output certifies this chosen epsilon and the explicit model, not an arbitrary sequence. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (9, 1/11, 3, 1/12, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-reciprocal-epsilon-witness · candidate
60. A Strict Reciprocal Bound for the Selected Epsilon
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=10, epsilon=1/12. 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.
View answer and explanation
(10, 1/12, 3, 1/13, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)
Retain the strict inequality when choosing N. For every two later indices the reciprocal terms lie between zero and the stated upper bound, so their distance is at most that bound. The output certifies this chosen epsilon and the explicit model, not an arbitrary sequence. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (10, 1/12, 3, 1/13, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-reciprocal-epsilon-witness · candidate
61. A Strict Reciprocal Bound for the Selected Epsilon
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=11, epsilon=1/13. 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.
View answer and explanation
(11, 1/13, 3, 1/14, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)
Retain the strict inequality when choosing N. For every two later indices the reciprocal terms lie between zero and the stated upper bound, so their distance is at most that bound. The output certifies this chosen epsilon and the explicit model, not an arbitrary sequence. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (11, 1/13, 3, 1/14, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-reciprocal-epsilon-witness · candidate
62. A Strict Reciprocal Bound for the Selected Epsilon
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=12, epsilon=1/14. 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.
View answer and explanation
(12, 1/14, 3, 1/15, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)
Retain the strict inequality when choosing N. For every two later indices the reciprocal terms lie between zero and the stated upper bound, so their distance is at most that bound. The output certifies this chosen epsilon and the explicit model, not an arbitrary sequence. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (12, 1/14, 3, 1/15, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-reciprocal-epsilon-witness · candidate
63. A Strict Reciprocal Bound for the Selected Epsilon
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=13, epsilon=1/15. 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.
View answer and explanation
(13, 1/15, 3, 1/16, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)
Retain the strict inequality when choosing N. For every two later indices the reciprocal terms lie between zero and the stated upper bound, so their distance is at most that bound. The output certifies this chosen epsilon and the explicit model, not an arbitrary sequence. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (13, 1/15, 3, 1/16, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-reciprocal-epsilon-witness · candidate
64. A Strict Reciprocal Bound for the Selected Epsilon
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=14, epsilon=1/16. 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.
View answer and explanation
(14, 1/16, 3, 1/17, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)
Retain the strict inequality when choosing N. For every two later indices the reciprocal terms lie between zero and the stated upper bound, so their distance is at most that bound. The output certifies this chosen epsilon and the explicit model, not an arbitrary sequence. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (14, 1/16, 3, 1/17, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-reciprocal-epsilon-witness · candidate
65. A Strict Reciprocal Bound for the Selected Epsilon
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=15, epsilon=1/17. 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.
View answer and explanation
(15, 1/17, 3, 1/18, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)
Retain the strict inequality when choosing N. For every two later indices the reciprocal terms lie between zero and the stated upper bound, so their distance is at most that bound. The output certifies this chosen epsilon and the explicit model, not an arbitrary sequence. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (15, 1/17, 3, 1/18, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-reciprocal-epsilon-witness · candidate
66. A Strict Reciprocal Bound for the Selected Epsilon
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=16, epsilon=1/18. 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.
View answer and explanation
(16, 1/18, 3, 1/19, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)
Retain the strict inequality when choosing N. For every two later indices the reciprocal terms lie between zero and the stated upper bound, so their distance is at most that bound. The output certifies this chosen epsilon and the explicit model, not an arbitrary sequence. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (16, 1/18, 3, 1/19, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-reciprocal-epsilon-witness · candidate
67. A Strict Reciprocal Bound for the Selected Epsilon
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=17, epsilon=1/19. 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.
View answer and explanation
(17, 1/19, 3, 1/20, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)
Retain the strict inequality when choosing N. For every two later indices the reciprocal terms lie between zero and the stated upper bound, so their distance is at most that bound. The output certifies this chosen epsilon and the explicit model, not an arbitrary sequence. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (17, 1/19, 3, 1/20, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-reciprocal-epsilon-witness · candidate
68. A Strict Reciprocal Bound for the Selected Epsilon
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=18, epsilon=1/20. 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.
View answer and explanation
(18, 1/20, 3, 1/21, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)
Retain the strict inequality when choosing N. For every two later indices the reciprocal terms lie between zero and the stated upper bound, so their distance is at most that bound. The output certifies this chosen epsilon and the explicit model, not an arbitrary sequence. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (18, 1/20, 3, 1/21, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-reciprocal-epsilon-witness · candidate
69. A Strict Reciprocal Bound for the Selected Epsilon
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=19, epsilon=1/21. 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.
View answer and explanation
(19, 1/21, 3, 1/22, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)
Retain the strict inequality when choosing N. For every two later indices the reciprocal terms lie between zero and the stated upper bound, so their distance is at most that bound. The output certifies this chosen epsilon and the explicit model, not an arbitrary sequence. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (19, 1/21, 3, 1/22, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-reciprocal-epsilon-witness · candidate
70. A Strict Reciprocal Bound for the Selected Epsilon
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=20, epsilon=1/22. 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.
View answer and explanation
(20, 1/22, 3, 1/23, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)
Retain the strict inequality when choosing N. For every two later indices the reciprocal terms lie between zero and the stated upper bound, so their distance is at most that bound. The output certifies this chosen epsilon and the explicit model, not an arbitrary sequence. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (20, 1/22, 3, 1/23, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-reciprocal-epsilon-witness · candidate
71. A Strict Reciprocal Bound for the Selected Epsilon
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=21, epsilon=1/23. 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.
View answer and explanation
(21, 1/23, 3, 1/24, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)
Retain the strict inequality when choosing N. For every two later indices the reciprocal terms lie between zero and the stated upper bound, so their distance is at most that bound. The output certifies this chosen epsilon and the explicit model, not an arbitrary sequence. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (21, 1/23, 3, 1/24, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-reciprocal-epsilon-witness · candidate
72. A Strict Reciprocal Bound for the Selected Epsilon
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=22, epsilon=1/24. 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.
View answer and explanation
(22, 1/24, 3, 1/25, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)
Retain the strict inequality when choosing N. For every two later indices the reciprocal terms lie between zero and the stated upper bound, so their distance is at most that bound. The output certifies this chosen epsilon and the explicit model, not an arbitrary sequence. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (22, 1/24, 3, 1/25, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-reciprocal-epsilon-witness · candidate
73. A Strict Reciprocal Bound for the Selected Epsilon
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=23, epsilon=1/25. 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.
View answer and explanation
(23, 1/25, 3, 1/26, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)
Retain the strict inequality when choosing N. For every two later indices the reciprocal terms lie between zero and the stated upper bound, so their distance is at most that bound. The output certifies this chosen epsilon and the explicit model, not an arbitrary sequence. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (23, 1/25, 3, 1/26, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-reciprocal-epsilon-witness · candidate
74. A Strict Reciprocal Bound for the Selected Epsilon
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=24, epsilon=1/26. 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.
View answer and explanation
(24, 1/26, 3, 1/27, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)
Retain the strict inequality when choosing N. For every two later indices the reciprocal terms lie between zero and the stated upper bound, so their distance is at most that bound. The output certifies this chosen epsilon and the explicit model, not an arbitrary sequence. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (24, 1/26, 3, 1/27, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-reciprocal-epsilon-witness · candidate
75. A Strict Reciprocal Bound for the Selected Epsilon
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=25, epsilon=1/27. 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.
View answer and explanation
(25, 1/27, 3, 1/28, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)
Retain the strict inequality when choosing N. For every two later indices the reciprocal terms lie between zero and the stated upper bound, so their distance is at most that bound. The output certifies this chosen epsilon and the explicit model, not an arbitrary sequence. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (25, 1/27, 3, 1/28, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-reciprocal-epsilon-witness · candidate
76. Adjacent Opposite Terms Refute Cauchy Behavior
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 sequence alternates between amplitude and its negative, with positive amplitude=1; arbitrary proposed cutoff N=1. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.
View answer and explanation
(1, 1, 2, 1, 2, Yes, No)
The adjacent terms at N and N+1 are both beyond the cutoff and have distance twice the amplitude, hence fail the strict epsilon bound. The same construction works for every N at one fixed positive epsilon: counterexample=Yes and cauchy=No. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 1, 2, 1, 2, Yes, No).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-alternating-counterexample · candidate
77. Adjacent Opposite Terms Refute Cauchy Behavior
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 sequence alternates between amplitude and its negative, with positive amplitude=2; arbitrary proposed cutoff N=3. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.
View answer and explanation
(2, 3, 4, 2, 4, Yes, No)
The adjacent terms at N and N+1 are both beyond the cutoff and have distance twice the amplitude, hence fail the strict epsilon bound. The same construction works for every N at one fixed positive epsilon: counterexample=Yes and cauchy=No. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 3, 4, 2, 4, Yes, No).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-alternating-counterexample · candidate
78. Adjacent Opposite Terms Refute Cauchy Behavior
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 sequence alternates between amplitude and its negative, with positive amplitude=3; arbitrary proposed cutoff N=5. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.
View answer and explanation
(3, 5, 6, 3, 6, Yes, No)
The adjacent terms at N and N+1 are both beyond the cutoff and have distance twice the amplitude, hence fail the strict epsilon bound. The same construction works for every N at one fixed positive epsilon: counterexample=Yes and cauchy=No. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3, 5, 6, 3, 6, Yes, No).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-alternating-counterexample · candidate
79. Adjacent Opposite Terms Refute Cauchy Behavior
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 sequence alternates between amplitude and its negative, with positive amplitude=4; arbitrary proposed cutoff N=7. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.
View answer and explanation
(4, 7, 8, 4, 8, Yes, No)
The adjacent terms at N and N+1 are both beyond the cutoff and have distance twice the amplitude, hence fail the strict epsilon bound. The same construction works for every N at one fixed positive epsilon: counterexample=Yes and cauchy=No. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4, 7, 8, 4, 8, Yes, No).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-alternating-counterexample · candidate
80. Adjacent Opposite Terms Refute Cauchy Behavior
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 sequence alternates between amplitude and its negative, with positive amplitude=5; arbitrary proposed cutoff N=9. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.
View answer and explanation
(5, 9, 10, 5, 10, Yes, No)
The adjacent terms at N and N+1 are both beyond the cutoff and have distance twice the amplitude, hence fail the strict epsilon bound. The same construction works for every N at one fixed positive epsilon: counterexample=Yes and cauchy=No. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5, 9, 10, 5, 10, Yes, No).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-alternating-counterexample · candidate
81. Adjacent Opposite Terms Refute Cauchy Behavior
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 sequence alternates between amplitude and its negative, with positive amplitude=6; arbitrary proposed cutoff N=11. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.
View answer and explanation
(6, 11, 12, 6, 12, Yes, No)
The adjacent terms at N and N+1 are both beyond the cutoff and have distance twice the amplitude, hence fail the strict epsilon bound. The same construction works for every N at one fixed positive epsilon: counterexample=Yes and cauchy=No. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, 11, 12, 6, 12, Yes, No).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-alternating-counterexample · candidate
82. Adjacent Opposite Terms Refute Cauchy Behavior
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 sequence alternates between amplitude and its negative, with positive amplitude=7; arbitrary proposed cutoff N=13. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.
View answer and explanation
(7, 13, 14, 7, 14, Yes, No)
The adjacent terms at N and N+1 are both beyond the cutoff and have distance twice the amplitude, hence fail the strict epsilon bound. The same construction works for every N at one fixed positive epsilon: counterexample=Yes and cauchy=No. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (7, 13, 14, 7, 14, Yes, No).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-alternating-counterexample · candidate
83. Adjacent Opposite Terms Refute Cauchy Behavior
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 sequence alternates between amplitude and its negative, with positive amplitude=8; arbitrary proposed cutoff N=15. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.
View answer and explanation
(8, 15, 16, 8, 16, Yes, No)
The adjacent terms at N and N+1 are both beyond the cutoff and have distance twice the amplitude, hence fail the strict epsilon bound. The same construction works for every N at one fixed positive epsilon: counterexample=Yes and cauchy=No. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (8, 15, 16, 8, 16, Yes, No).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-alternating-counterexample · candidate
84. Adjacent Opposite Terms Refute Cauchy Behavior
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 sequence alternates between amplitude and its negative, with positive amplitude=9; arbitrary proposed cutoff N=17. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.
View answer and explanation
(9, 17, 18, 9, 18, Yes, No)
The adjacent terms at N and N+1 are both beyond the cutoff and have distance twice the amplitude, hence fail the strict epsilon bound. The same construction works for every N at one fixed positive epsilon: counterexample=Yes and cauchy=No. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (9, 17, 18, 9, 18, Yes, No).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-alternating-counterexample · candidate
85. Adjacent Opposite Terms Refute Cauchy Behavior
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 sequence alternates between amplitude and its negative, with positive amplitude=10; arbitrary proposed cutoff N=19. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.
View answer and explanation
(10, 19, 20, 10, 20, Yes, No)
The adjacent terms at N and N+1 are both beyond the cutoff and have distance twice the amplitude, hence fail the strict epsilon bound. The same construction works for every N at one fixed positive epsilon: counterexample=Yes and cauchy=No. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (10, 19, 20, 10, 20, Yes, No).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-alternating-counterexample · candidate
86. Adjacent Opposite Terms Refute Cauchy Behavior
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 sequence alternates between amplitude and its negative, with positive amplitude=11; arbitrary proposed cutoff N=21. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.
View answer and explanation
(11, 21, 22, 11, 22, Yes, No)
The adjacent terms at N and N+1 are both beyond the cutoff and have distance twice the amplitude, hence fail the strict epsilon bound. The same construction works for every N at one fixed positive epsilon: counterexample=Yes and cauchy=No. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (11, 21, 22, 11, 22, Yes, No).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-alternating-counterexample · candidate
87. Adjacent Opposite Terms Refute Cauchy Behavior
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 sequence alternates between amplitude and its negative, with positive amplitude=12; arbitrary proposed cutoff N=23. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.
View answer and explanation
(12, 23, 24, 12, 24, Yes, No)
The adjacent terms at N and N+1 are both beyond the cutoff and have distance twice the amplitude, hence fail the strict epsilon bound. The same construction works for every N at one fixed positive epsilon: counterexample=Yes and cauchy=No. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (12, 23, 24, 12, 24, Yes, No).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-alternating-counterexample · candidate
88. Adjacent Opposite Terms Refute Cauchy Behavior
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 sequence alternates between amplitude and its negative, with positive amplitude=13; arbitrary proposed cutoff N=25. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.
View answer and explanation
(13, 25, 26, 13, 26, Yes, No)
The adjacent terms at N and N+1 are both beyond the cutoff and have distance twice the amplitude, hence fail the strict epsilon bound. The same construction works for every N at one fixed positive epsilon: counterexample=Yes and cauchy=No. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (13, 25, 26, 13, 26, Yes, No).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-alternating-counterexample · candidate
89. Adjacent Opposite Terms Refute Cauchy Behavior
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 sequence alternates between amplitude and its negative, with positive amplitude=14; arbitrary proposed cutoff N=27. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.
View answer and explanation
(14, 27, 28, 14, 28, Yes, No)
The adjacent terms at N and N+1 are both beyond the cutoff and have distance twice the amplitude, hence fail the strict epsilon bound. The same construction works for every N at one fixed positive epsilon: counterexample=Yes and cauchy=No. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (14, 27, 28, 14, 28, Yes, No).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-alternating-counterexample · candidate
90. Adjacent Opposite Terms Refute Cauchy Behavior
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 sequence alternates between amplitude and its negative, with positive amplitude=15; arbitrary proposed cutoff N=29. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.
View answer and explanation
(15, 29, 30, 15, 30, Yes, No)
The adjacent terms at N and N+1 are both beyond the cutoff and have distance twice the amplitude, hence fail the strict epsilon bound. The same construction works for every N at one fixed positive epsilon: counterexample=Yes and cauchy=No. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (15, 29, 30, 15, 30, Yes, No).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-alternating-counterexample · candidate
91. Adjacent Opposite Terms Refute Cauchy Behavior
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 sequence alternates between amplitude and its negative, with positive amplitude=16; arbitrary proposed cutoff N=31. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.
View answer and explanation
(16, 31, 32, 16, 32, Yes, No)
The adjacent terms at N and N+1 are both beyond the cutoff and have distance twice the amplitude, hence fail the strict epsilon bound. The same construction works for every N at one fixed positive epsilon: counterexample=Yes and cauchy=No. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (16, 31, 32, 16, 32, Yes, No).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-alternating-counterexample · candidate
92. Adjacent Opposite Terms Refute Cauchy Behavior
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 sequence alternates between amplitude and its negative, with positive amplitude=17; arbitrary proposed cutoff N=33. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.
View answer and explanation
(17, 33, 34, 17, 34, Yes, No)
The adjacent terms at N and N+1 are both beyond the cutoff and have distance twice the amplitude, hence fail the strict epsilon bound. The same construction works for every N at one fixed positive epsilon: counterexample=Yes and cauchy=No. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (17, 33, 34, 17, 34, Yes, No).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-alternating-counterexample · candidate
93. Adjacent Opposite Terms Refute Cauchy Behavior
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 sequence alternates between amplitude and its negative, with positive amplitude=18; arbitrary proposed cutoff N=35. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.
View answer and explanation
(18, 35, 36, 18, 36, Yes, No)
The adjacent terms at N and N+1 are both beyond the cutoff and have distance twice the amplitude, hence fail the strict epsilon bound. The same construction works for every N at one fixed positive epsilon: counterexample=Yes and cauchy=No. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (18, 35, 36, 18, 36, Yes, No).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-alternating-counterexample · candidate
94. Adjacent Opposite Terms Refute Cauchy Behavior
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 sequence alternates between amplitude and its negative, with positive amplitude=19; arbitrary proposed cutoff N=37. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.
View answer and explanation
(19, 37, 38, 19, 38, Yes, No)
The adjacent terms at N and N+1 are both beyond the cutoff and have distance twice the amplitude, hence fail the strict epsilon bound. The same construction works for every N at one fixed positive epsilon: counterexample=Yes and cauchy=No. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (19, 37, 38, 19, 38, Yes, No).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-alternating-counterexample · candidate
95. Adjacent Opposite Terms Refute Cauchy Behavior
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 sequence alternates between amplitude and its negative, with positive amplitude=20; arbitrary proposed cutoff N=39. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.
View answer and explanation
(20, 39, 40, 20, 40, Yes, No)
The adjacent terms at N and N+1 are both beyond the cutoff and have distance twice the amplitude, hence fail the strict epsilon bound. The same construction works for every N at one fixed positive epsilon: counterexample=Yes and cauchy=No. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (20, 39, 40, 20, 40, Yes, No).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-alternating-counterexample · candidate
96. Adjacent Opposite Terms Refute Cauchy Behavior
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 sequence alternates between amplitude and its negative, with positive amplitude=21; arbitrary proposed cutoff N=41. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.
View answer and explanation
(21, 41, 42, 21, 42, Yes, No)
The adjacent terms at N and N+1 are both beyond the cutoff and have distance twice the amplitude, hence fail the strict epsilon bound. The same construction works for every N at one fixed positive epsilon: counterexample=Yes and cauchy=No. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (21, 41, 42, 21, 42, Yes, No).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-alternating-counterexample · candidate
97. Adjacent Opposite Terms Refute Cauchy Behavior
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 sequence alternates between amplitude and its negative, with positive amplitude=22; arbitrary proposed cutoff N=43. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.
View answer and explanation
(22, 43, 44, 22, 44, Yes, No)
The adjacent terms at N and N+1 are both beyond the cutoff and have distance twice the amplitude, hence fail the strict epsilon bound. The same construction works for every N at one fixed positive epsilon: counterexample=Yes and cauchy=No. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (22, 43, 44, 22, 44, Yes, No).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-alternating-counterexample · candidate
98. Adjacent Opposite Terms Refute Cauchy Behavior
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 sequence alternates between amplitude and its negative, with positive amplitude=23; arbitrary proposed cutoff N=45. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.
View answer and explanation
(23, 45, 46, 23, 46, Yes, No)
The adjacent terms at N and N+1 are both beyond the cutoff and have distance twice the amplitude, hence fail the strict epsilon bound. The same construction works for every N at one fixed positive epsilon: counterexample=Yes and cauchy=No. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (23, 45, 46, 23, 46, Yes, No).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-alternating-counterexample · candidate
99. Adjacent Opposite Terms Refute Cauchy Behavior
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 sequence alternates between amplitude and its negative, with positive amplitude=24; arbitrary proposed cutoff N=47. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.
View answer and explanation
(24, 47, 48, 24, 48, Yes, No)
The adjacent terms at N and N+1 are both beyond the cutoff and have distance twice the amplitude, hence fail the strict epsilon bound. The same construction works for every N at one fixed positive epsilon: counterexample=Yes and cauchy=No. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (24, 47, 48, 24, 48, Yes, No).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-alternating-counterexample · candidate
100. Adjacent Opposite Terms Refute Cauchy Behavior
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 sequence alternates between amplitude and its negative, with positive amplitude=25; arbitrary proposed cutoff N=49. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.
View answer and explanation
(25, 49, 50, 25, 50, Yes, No)
The adjacent terms at N and N+1 are both beyond the cutoff and have distance twice the amplitude, hence fail the strict epsilon bound. The same construction works for every N at one fixed positive epsilon: counterexample=Yes and cauchy=No. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (25, 49, 50, 25, 50, Yes, No).
Reviewed step-by-step derivation
Cauchy sequence · qf-mod26-cauchy-alternating-counterexample · candidate
Canonical question JSONL · English JSONL · Validation report