# Question 21436: Parity Subsequences Need a Strict All-Later Epsilon Witness

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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.

R0041-21436 · q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0061



## Original question and givens

Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=11/12, c=11>0, shift=12≥1, parity=0 and epsilon=1/20>0. The infinite index rule is k(j)=2j+parity for every j≥0. Give (parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate).


- G001 — Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=11/12, c=11>0, shift=12≥1, parity=0 and epsilon=1/20>0. The infinite index rule is k(j)=2j+parity for every j≥0. Give (parity, index_rule, limit, epsilon, J, value_at_k_J, 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-bolzano-weierstrass-theorem-real](https://ocme.evemisslab.com/en/mko/mko-bolzano-weierstrass-theorem-real/)





## Step-by-step derivation

S001 · impliesBefore

source operation=bw-oscillatory-tail;original snapshot and mathematics retained

After

explicit ordered source output fields=[parity, index_rule, limit, epsilon, J, value_at_k_J, 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 a_n=11/12+(-1)^n*(11)/(n+12) for EVERY n>=0;parity=0

After

k(j)=2j+0 for ALL j>=0;k(j+1)-k(j)=2>0;selected sign=1;whole sequence bounded by |L|+c/shift

Conditions: none

The parity rule is an infinite strictly increasing index map.Selected signs are fixed and errors equal c/(2j+parity+shift).The whole sequence is bounded;its monotonicity is not a premise for this subsequence certificate.

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





S003 · calculationBefore

bw c over epsilon := div(11, 1/20)

After

bw c over epsilon = 220

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, S002





S004 · calculationBefore

bw index multiple := mul(2, 105)

After

bw index multiple = 210

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

bw actual source index := sum(0, 210)

After

bw actual source index = 210

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

bw witness denominator := sum(210, 12)

After

bw witness denominator = 222

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

bw witness distance := div(11, 222)

After

bw witness distance = 11/222

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

bw predecessor distance := div(11, 220)

After

bw predecessor distance = 1/20

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 · impliesBefore

bw:c=11,shift=12,epsilon=1/20;index(q)=2q+0;c/epsilon=220;floor=220

After

least nonnegative witness=105;actual source index=210;distance=11/222<epsilon;predecessor distance=1/20;no predecessor evaluated when witness0

Conditions: none

Strict distance<epsilon means the denominator exceeds c/epsilon,so its integer threshold is floor(c/epsilon)+1.Round the affine index upward and clamp only the witness to0.Check the predecessor only when positive;equality still fails.

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





S010 · impliesBefore

bw:positive c=11,epsilon=1/20,shift=12,stride=2,offset=0;least witness=105

After

for EVERY integer q>=105,denominator=2q+0+12>=witness denominator>0,so 0<distance<=(11/222)<epsilon

Conditions: none

This is an algebraic inequality for every later index,not convergence guessed from one evaluated term.Positive increasing denominators and a fixed positive numerator control the whole tail;the predecessor check proves minimality only.

Step identifiers and dependenciesRule: tail_all_later_quantified_from_positive_denominator_growth · Uses: G001, S009





S011 · calculationBefore

bw at selected k(J) denominator := sum(210, 12)

After

bw at selected k(J) denominator = 222

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, S010





S012 · calculationBefore

bw at selected k(J) unsigned tail := div(11, 222)

After

bw at selected k(J) unsigned tail = 11/222

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, S011





S013 · calculationBefore

bw at selected k(J) signed value := sum(11/12, 11/222)

After

bw at selected k(J) signed value = 143/148

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, S012









## Reverse checks


- S013 · premise_and_result_audit — Recompute bw at selected k(J) signed value with exact rational arithmetic, not rounded decimals.
- S012 · premise_and_result_audit — Recompute bw at selected k(J) unsigned tail with exact rational arithmetic, not rounded decimals.
- S011 · premise_and_result_audit — Recompute bw at selected k(J) denominator with exact rational arithmetic, not rounded decimals.
- S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S008 · premise_and_result_audit — Recompute bw predecessor distance with exact rational arithmetic, not rounded decimals.
- S007 · premise_and_result_audit — Recompute bw witness distance with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Recompute bw witness denominator with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Recompute bw actual source index with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute bw index multiple with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute bw c over epsilon with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(0, k(j)=2j+0, 11/12, 1/20, 105, 143/148, Yes, forall_j_ge_J_symbolic_subsequence_tail)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 8b2aa74b1877b1b4127637d58cb5023378985cc021d828c0a26f58d7f080499e
- candidate record SHA-256: e62b68c10a7fa9756a559eef2003f2141d3c8e2c4e3c9d9be9322cbf4040927c
- semantic review receipt SHA-256: 8f134403487694af4c70d2e00ec84bab542043ef4b3c54517c04f3c409c0262a
- question-first witness SHA-256: ad772845088ed8e99f87f90d4264b05ea97be35775305a02011af5ee35f0f4ce
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0061.json)