# Question 21045: A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries [Return to catalogue page 106](https://ocme.evemisslab.com/en/questions/page/106/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations/#q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0070) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0041-21045 · q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0070 ## Original question and givens Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=20/21, positive c=20 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=decreasing, positive epsilon=1/29. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate). - G001 — Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=20/21, positive c=20 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=decreasing, positive epsilon=1/29. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-monotone-sequence-convergence-theorem](https://ocme.evemisslab.com/en/mko/mko-monotone-sequence-convergence-theorem/) ## Step-by-step derivation S001 · impliesBefore source operation=monoseq-epsilon-witness;original snapshot and mathematics retained After explicit ordered source output fields=[direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate] Conditions: none Retain the historical interface marker while listing ordered output fields without JavaScript.Fields and tags are not extra mathematical evidence;fixed No markers retain their claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore ORIGINAL direction=decreasing,L=20/21,c=20>0,shift=2>=1;WHOLE a_n=L+(1)c/(n+shift) for ALL n>=0 After for EVERY n>=0,a_(n+1)-a_n=(-1)*(20)/((n+2)*(n+1+2));positive denominator;all a_n strictly above L=20/21 Conditions: none Subtract adjacent symbolic terms using a common denominator.Positivity fixes the sign and gives upper bound L in the increasing case or lower bound L in the decreasing case.This holds for all n,allows negative L,and is not a three-term sample proof. Step identifiers and dependenciesRule: monoseq_whole_symbolic_monotonicity_and_matching_bound · Uses: G001, S001 S003 · impliesBefore WHOLE decreasing law with ORIGINAL L=20/21,c=20,shift=2 After for ANY epsilon>0 choose natural N with N+2>20/epsilon;ALL n>=N have |a_n-L|0 Conditions: none The Archimedean choice controls every later term.Any smaller upper bound or larger lower bound is broken by a term sufficiently close to L,so L is the sup/inf.In the general monotone theorem,sup−epsilon is not an upper bound,which supplies a term;monotonicity then traps the tail.The decreasing proof is dual. Step identifiers and dependenciesRule: monoseq_sup_inf_limit_proof_and_nonattainment · Uses: G001, S002 S004 · calculationBefore monoseq c over epsilon := div(20, 1/29) After monoseq c over epsilon = 580 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore monoseq index multiple := mul(1, 579) After monoseq index multiple = 579 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore monoseq actual source index := sum(0, 579) After monoseq actual source index = 579 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore monoseq witness denominator := sum(579, 2) After monoseq witness denominator = 581 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore monoseq witness distance := div(20, 581) After monoseq witness distance = 20/581 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore monoseq predecessor distance := div(20, 580) After monoseq predecessor distance = 1/29 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore monoseq:c=20,shift=2,epsilon=1/29;index(q)=1q+0;c/epsilon=580;floor=580 After least nonnegative witness=579;actual source index=579;distance=20/581=579,denominator=1q+0+2>=witness denominator>0,so 0