# Whole Bounded Sequences, Infinite Index Maps and Strict Subsequence Tails

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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-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. A Whole Declared Periodic Sequence Has an Exact Absolute Bound

Use the whole infinite real periodic sequence with nonempty period_values=[1, -2, 3/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate).

View answer and explanation([1, -2, 3/2], 2, Yes, whole_infinite_periodic_sequence_bounded)

Take the largest absolute value in the complete period as a bound for every term of the infinite sequence. This finite periodic certificate does not prove the theorem for arbitrary sequences. The exact result, retaining the requested order, is ([1, -2, 3/2], 2, Yes, whole_infinite_periodic_sequence_bounded).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0001/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-boundedness · candidate





## 2. A Whole Declared Periodic Sequence Has an Exact Absolute Bound

Use the whole infinite real periodic sequence with nonempty period_values=[2, -3, 4/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate).

View answer and explanation([2, -3, 2], 3, Yes, whole_infinite_periodic_sequence_bounded)

Take the largest absolute value in the complete period as a bound for every term of the infinite sequence. This finite periodic certificate does not prove the theorem for arbitrary sequences. The exact result, retaining the requested order, is ([2, -3, 2], 3, Yes, whole_infinite_periodic_sequence_bounded).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0002/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-boundedness · candidate





## 3. A Whole Declared Periodic Sequence Has an Exact Absolute Bound

Use the whole infinite real periodic sequence with nonempty period_values=[3, -4, 5/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate).

View answer and explanation([3, -4, 5/2], 4, Yes, whole_infinite_periodic_sequence_bounded)

Take the largest absolute value in the complete period as a bound for every term of the infinite sequence. This finite periodic certificate does not prove the theorem for arbitrary sequences. The exact result, retaining the requested order, is ([3, -4, 5/2], 4, Yes, whole_infinite_periodic_sequence_bounded).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0003/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-boundedness · candidate





## 4. A Whole Declared Periodic Sequence Has an Exact Absolute Bound

Use the whole infinite real periodic sequence with nonempty period_values=[4, -5, 6/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate).

View answer and explanation([4, -5, 3], 5, Yes, whole_infinite_periodic_sequence_bounded)

Take the largest absolute value in the complete period as a bound for every term of the infinite sequence. This finite periodic certificate does not prove the theorem for arbitrary sequences. The exact result, retaining the requested order, is ([4, -5, 3], 5, Yes, whole_infinite_periodic_sequence_bounded).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0004/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-boundedness · candidate





## 5. A Whole Declared Periodic Sequence Has an Exact Absolute Bound

Use the whole infinite real periodic sequence with nonempty period_values=[5, -6, 7/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate).

View answer and explanation([5, -6, 7/2], 6, Yes, whole_infinite_periodic_sequence_bounded)

Take the largest absolute value in the complete period as a bound for every term of the infinite sequence. This finite periodic certificate does not prove the theorem for arbitrary sequences. The exact result, retaining the requested order, is ([5, -6, 7/2], 6, Yes, whole_infinite_periodic_sequence_bounded).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0005/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-boundedness · candidate





## 6. A Whole Declared Periodic Sequence Has an Exact Absolute Bound

Use the whole infinite real periodic sequence with nonempty period_values=[6, -7, 8/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate).

View answer and explanation([6, -7, 4], 7, Yes, whole_infinite_periodic_sequence_bounded)

Take the largest absolute value in the complete period as a bound for every term of the infinite sequence. This finite periodic certificate does not prove the theorem for arbitrary sequences. The exact result, retaining the requested order, is ([6, -7, 4], 7, Yes, whole_infinite_periodic_sequence_bounded).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0006/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-boundedness · candidate





## 7. A Whole Declared Periodic Sequence Has an Exact Absolute Bound

Use the whole infinite real periodic sequence with nonempty period_values=[7, -8, 9/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate).

View answer and explanation([7, -8, 9/2], 8, Yes, whole_infinite_periodic_sequence_bounded)

Take the largest absolute value in the complete period as a bound for every term of the infinite sequence. This finite periodic certificate does not prove the theorem for arbitrary sequences. The exact result, retaining the requested order, is ([7, -8, 9/2], 8, Yes, whole_infinite_periodic_sequence_bounded).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0007/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-boundedness · candidate





## 8. A Whole Declared Periodic Sequence Has an Exact Absolute Bound

Use the whole infinite real periodic sequence with nonempty period_values=[8, -9, 10/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate).

View answer and explanation([8, -9, 5], 9, Yes, whole_infinite_periodic_sequence_bounded)

Take the largest absolute value in the complete period as a bound for every term of the infinite sequence. This finite periodic certificate does not prove the theorem for arbitrary sequences. The exact result, retaining the requested order, is ([8, -9, 5], 9, Yes, whole_infinite_periodic_sequence_bounded).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0008/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-boundedness · candidate





## 9. A Whole Declared Periodic Sequence Has an Exact Absolute Bound

Use the whole infinite real periodic sequence with nonempty period_values=[9, -10, 11/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate).

View answer and explanation([9, -10, 11/2], 10, Yes, whole_infinite_periodic_sequence_bounded)

Take the largest absolute value in the complete period as a bound for every term of the infinite sequence. This finite periodic certificate does not prove the theorem for arbitrary sequences. The exact result, retaining the requested order, is ([9, -10, 11/2], 10, Yes, whole_infinite_periodic_sequence_bounded).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0009/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-boundedness · candidate





## 10. A Whole Declared Periodic Sequence Has an Exact Absolute Bound

Use the whole infinite real periodic sequence with nonempty period_values=[10, -11, 12/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate).

View answer and explanation([10, -11, 6], 11, Yes, whole_infinite_periodic_sequence_bounded)

Take the largest absolute value in the complete period as a bound for every term of the infinite sequence. This finite periodic certificate does not prove the theorem for arbitrary sequences. The exact result, retaining the requested order, is ([10, -11, 6], 11, Yes, whole_infinite_periodic_sequence_bounded).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0010/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-boundedness · candidate





## 11. A Whole Declared Periodic Sequence Has an Exact Absolute Bound

Use the whole infinite real periodic sequence with nonempty period_values=[11, -12, 13/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate).

View answer and explanation([11, -12, 13/2], 12, Yes, whole_infinite_periodic_sequence_bounded)

Take the largest absolute value in the complete period as a bound for every term of the infinite sequence. This finite periodic certificate does not prove the theorem for arbitrary sequences. The exact result, retaining the requested order, is ([11, -12, 13/2], 12, Yes, whole_infinite_periodic_sequence_bounded).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0011/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-boundedness · candidate





## 12. A Whole Declared Periodic Sequence Has an Exact Absolute Bound

Use the whole infinite real periodic sequence with nonempty period_values=[12, -13, 14/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate).

View answer and explanation([12, -13, 7], 13, Yes, whole_infinite_periodic_sequence_bounded)

Take the largest absolute value in the complete period as a bound for every term of the infinite sequence. This finite periodic certificate does not prove the theorem for arbitrary sequences. The exact result, retaining the requested order, is ([12, -13, 7], 13, Yes, whole_infinite_periodic_sequence_bounded).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0012/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-boundedness · candidate





## 13. A Whole Declared Periodic Sequence Has an Exact Absolute Bound

Use the whole infinite real periodic sequence with nonempty period_values=[13, -14, 15/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate).

View answer and explanation([13, -14, 15/2], 14, Yes, whole_infinite_periodic_sequence_bounded)

Take the largest absolute value in the complete period as a bound for every term of the infinite sequence. This finite periodic certificate does not prove the theorem for arbitrary sequences. The exact result, retaining the requested order, is ([13, -14, 15/2], 14, Yes, whole_infinite_periodic_sequence_bounded).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0013/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-boundedness · candidate





## 14. A Whole Declared Periodic Sequence Has an Exact Absolute Bound

Use the whole infinite real periodic sequence with nonempty period_values=[14, -15, 16/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate).

View answer and explanation([14, -15, 8], 15, Yes, whole_infinite_periodic_sequence_bounded)

Take the largest absolute value in the complete period as a bound for every term of the infinite sequence. This finite periodic certificate does not prove the theorem for arbitrary sequences. The exact result, retaining the requested order, is ([14, -15, 8], 15, Yes, whole_infinite_periodic_sequence_bounded).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0014/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-boundedness · candidate





## 15. A Whole Declared Periodic Sequence Has an Exact Absolute Bound

Use the whole infinite real periodic sequence with nonempty period_values=[15, -16, 17/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate).

View answer and explanation([15, -16, 17/2], 16, Yes, whole_infinite_periodic_sequence_bounded)

Take the largest absolute value in the complete period as a bound for every term of the infinite sequence. This finite periodic certificate does not prove the theorem for arbitrary sequences. The exact result, retaining the requested order, is ([15, -16, 17/2], 16, Yes, whole_infinite_periodic_sequence_bounded).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0015/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-boundedness · candidate





## 16. A Whole Declared Periodic Sequence Has an Exact Absolute Bound

Use the whole infinite real periodic sequence with nonempty period_values=[16, -17, 18/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate).

View answer and explanation([16, -17, 9], 17, Yes, whole_infinite_periodic_sequence_bounded)

Take the largest absolute value in the complete period as a bound for every term of the infinite sequence. This finite periodic certificate does not prove the theorem for arbitrary sequences. The exact result, retaining the requested order, is ([16, -17, 9], 17, Yes, whole_infinite_periodic_sequence_bounded).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0016/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-boundedness · candidate





## 17. A Whole Declared Periodic Sequence Has an Exact Absolute Bound

Use the whole infinite real periodic sequence with nonempty period_values=[17, -18, 19/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate).

View answer and explanation([17, -18, 19/2], 18, Yes, whole_infinite_periodic_sequence_bounded)

Take the largest absolute value in the complete period as a bound for every term of the infinite sequence. This finite periodic certificate does not prove the theorem for arbitrary sequences. The exact result, retaining the requested order, is ([17, -18, 19/2], 18, Yes, whole_infinite_periodic_sequence_bounded).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0017/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-boundedness · candidate





## 18. A Whole Declared Periodic Sequence Has an Exact Absolute Bound

Use the whole infinite real periodic sequence with nonempty period_values=[18, -19, 20/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate).

View answer and explanation([18, -19, 10], 19, Yes, whole_infinite_periodic_sequence_bounded)

Take the largest absolute value in the complete period as a bound for every term of the infinite sequence. This finite periodic certificate does not prove the theorem for arbitrary sequences. The exact result, retaining the requested order, is ([18, -19, 10], 19, Yes, whole_infinite_periodic_sequence_bounded).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0018/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-boundedness · candidate





## 19. A Whole Declared Periodic Sequence Has an Exact Absolute Bound

Use the whole infinite real periodic sequence with nonempty period_values=[19, -20, 21/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate).

View answer and explanation([19, -20, 21/2], 20, Yes, whole_infinite_periodic_sequence_bounded)

Take the largest absolute value in the complete period as a bound for every term of the infinite sequence. This finite periodic certificate does not prove the theorem for arbitrary sequences. The exact result, retaining the requested order, is ([19, -20, 21/2], 20, Yes, whole_infinite_periodic_sequence_bounded).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0019/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-boundedness · candidate





## 20. A Whole Declared Periodic Sequence Has an Exact Absolute Bound

Use the whole infinite real periodic sequence with nonempty period_values=[20, -21, 22/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate).

View answer and explanation([20, -21, 11], 21, Yes, whole_infinite_periodic_sequence_bounded)

Take the largest absolute value in the complete period as a bound for every term of the infinite sequence. This finite periodic certificate does not prove the theorem for arbitrary sequences. The exact result, retaining the requested order, is ([20, -21, 11], 21, Yes, whole_infinite_periodic_sequence_bounded).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0020/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-boundedness · candidate





## 21. A Whole Declared Periodic Sequence Has an Exact Absolute Bound

Use the whole infinite real periodic sequence with nonempty period_values=[21, -22, 23/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate).

View answer and explanation([21, -22, 23/2], 22, Yes, whole_infinite_periodic_sequence_bounded)

Take the largest absolute value in the complete period as a bound for every term of the infinite sequence. This finite periodic certificate does not prove the theorem for arbitrary sequences. The exact result, retaining the requested order, is ([21, -22, 23/2], 22, Yes, whole_infinite_periodic_sequence_bounded).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0021/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-boundedness · candidate





## 22. A Whole Declared Periodic Sequence Has an Exact Absolute Bound

Use the whole infinite real periodic sequence with nonempty period_values=[22, -23, 24/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate).

View answer and explanation([22, -23, 12], 23, Yes, whole_infinite_periodic_sequence_bounded)

Take the largest absolute value in the complete period as a bound for every term of the infinite sequence. This finite periodic certificate does not prove the theorem for arbitrary sequences. The exact result, retaining the requested order, is ([22, -23, 12], 23, Yes, whole_infinite_periodic_sequence_bounded).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0022/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-boundedness · candidate





## 23. A Whole Declared Periodic Sequence Has an Exact Absolute Bound

Use the whole infinite real periodic sequence with nonempty period_values=[23, -24, 25/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate).

View answer and explanation([23, -24, 25/2], 24, Yes, whole_infinite_periodic_sequence_bounded)

Take the largest absolute value in the complete period as a bound for every term of the infinite sequence. This finite periodic certificate does not prove the theorem for arbitrary sequences. The exact result, retaining the requested order, is ([23, -24, 25/2], 24, Yes, whole_infinite_periodic_sequence_bounded).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0023/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-boundedness · candidate





## 24. A Whole Declared Periodic Sequence Has an Exact Absolute Bound

Use the whole infinite real periodic sequence with nonempty period_values=[24, -25, 26/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate).

View answer and explanation([24, -25, 13], 25, Yes, whole_infinite_periodic_sequence_bounded)

Take the largest absolute value in the complete period as a bound for every term of the infinite sequence. This finite periodic certificate does not prove the theorem for arbitrary sequences. The exact result, retaining the requested order, is ([24, -25, 13], 25, Yes, whole_infinite_periodic_sequence_bounded).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0024/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-boundedness · candidate





## 25. A Whole Declared Periodic Sequence Has an Exact Absolute Bound

Use the whole infinite real periodic sequence with nonempty period_values=[25, -26, 27/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Give (period_values, absolute_bound, whole_sequence_bounded, certificate).

View answer and explanation([25, -26, 27/2], 26, Yes, whole_infinite_periodic_sequence_bounded)

Take the largest absolute value in the complete period as a bound for every term of the infinite sequence. This finite periodic certificate does not prove the theorem for arbitrary sequences. The exact result, retaining the requested order, is ([25, -26, 27/2], 26, Yes, whole_infinite_periodic_sequence_bounded).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0025/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-boundedness · candidate





## 26. A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence

Use the whole infinite real periodic sequence with nonempty period_values=[1, -2, 3/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=0 and subsequence parameter j=1≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate).

View answer and explanation(0, 1, 3, 1, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

With period length t, k(j)=residue+t*j is strictly increasing on all nonnegative j, and every selected value is the same period entry. The displayed j only evaluates this infinite index law. The exact result, retaining the requested order, is (0, 1, 3, 1, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0026/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-subsequence · candidate





## 27. A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence

Use the whole infinite real periodic sequence with nonempty period_values=[2, -3, 4/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=1 and subsequence parameter j=2≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate).

View answer and explanation(1, 2, 7, -3, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

With period length t, k(j)=residue+t*j is strictly increasing on all nonnegative j, and every selected value is the same period entry. The displayed j only evaluates this infinite index law. The exact result, retaining the requested order, is (1, 2, 7, -3, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0027/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-subsequence · candidate





## 28. A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence

Use the whole infinite real periodic sequence with nonempty period_values=[3, -4, 5/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=2 and subsequence parameter j=3≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate).

View answer and explanation(2, 3, 11, 5/2, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

With period length t, k(j)=residue+t*j is strictly increasing on all nonnegative j, and every selected value is the same period entry. The displayed j only evaluates this infinite index law. The exact result, retaining the requested order, is (2, 3, 11, 5/2, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0028/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-subsequence · candidate





## 29. A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence

Use the whole infinite real periodic sequence with nonempty period_values=[4, -5, 6/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=0 and subsequence parameter j=4≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate).

View answer and explanation(0, 4, 12, 4, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

With period length t, k(j)=residue+t*j is strictly increasing on all nonnegative j, and every selected value is the same period entry. The displayed j only evaluates this infinite index law. The exact result, retaining the requested order, is (0, 4, 12, 4, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0029/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-subsequence · candidate





## 30. A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence

Use the whole infinite real periodic sequence with nonempty period_values=[5, -6, 7/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=1 and subsequence parameter j=5≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate).

View answer and explanation(1, 5, 16, -6, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

With period length t, k(j)=residue+t*j is strictly increasing on all nonnegative j, and every selected value is the same period entry. The displayed j only evaluates this infinite index law. The exact result, retaining the requested order, is (1, 5, 16, -6, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0030/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-subsequence · candidate





## 31. A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence

Use the whole infinite real periodic sequence with nonempty period_values=[6, -7, 8/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=2 and subsequence parameter j=6≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate).

View answer and explanation(2, 6, 20, 4, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

With period length t, k(j)=residue+t*j is strictly increasing on all nonnegative j, and every selected value is the same period entry. The displayed j only evaluates this infinite index law. The exact result, retaining the requested order, is (2, 6, 20, 4, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0031/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-subsequence · candidate





## 32. A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence

Use the whole infinite real periodic sequence with nonempty period_values=[7, -8, 9/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=0 and subsequence parameter j=7≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate).

View answer and explanation(0, 7, 21, 7, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

With period length t, k(j)=residue+t*j is strictly increasing on all nonnegative j, and every selected value is the same period entry. The displayed j only evaluates this infinite index law. The exact result, retaining the requested order, is (0, 7, 21, 7, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0032/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-subsequence · candidate





## 33. A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence

Use the whole infinite real periodic sequence with nonempty period_values=[8, -9, 10/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=1 and subsequence parameter j=8≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate).

View answer and explanation(1, 8, 25, -9, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

With period length t, k(j)=residue+t*j is strictly increasing on all nonnegative j, and every selected value is the same period entry. The displayed j only evaluates this infinite index law. The exact result, retaining the requested order, is (1, 8, 25, -9, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0033/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-subsequence · candidate





## 34. A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence

Use the whole infinite real periodic sequence with nonempty period_values=[9, -10, 11/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=2 and subsequence parameter j=9≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate).

View answer and explanation(2, 9, 29, 11/2, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

With period length t, k(j)=residue+t*j is strictly increasing on all nonnegative j, and every selected value is the same period entry. The displayed j only evaluates this infinite index law. The exact result, retaining the requested order, is (2, 9, 29, 11/2, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0034/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-subsequence · candidate





## 35. A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence

Use the whole infinite real periodic sequence with nonempty period_values=[10, -11, 12/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=0 and subsequence parameter j=10≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate).

View answer and explanation(0, 10, 30, 10, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

With period length t, k(j)=residue+t*j is strictly increasing on all nonnegative j, and every selected value is the same period entry. The displayed j only evaluates this infinite index law. The exact result, retaining the requested order, is (0, 10, 30, 10, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0035/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-subsequence · candidate





## 36. A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence

Use the whole infinite real periodic sequence with nonempty period_values=[11, -12, 13/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=1 and subsequence parameter j=11≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate).

View answer and explanation(1, 11, 34, -12, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

With period length t, k(j)=residue+t*j is strictly increasing on all nonnegative j, and every selected value is the same period entry. The displayed j only evaluates this infinite index law. The exact result, retaining the requested order, is (1, 11, 34, -12, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0036/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-subsequence · candidate





## 37. A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence

Use the whole infinite real periodic sequence with nonempty period_values=[12, -13, 14/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=2 and subsequence parameter j=12≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate).

View answer and explanation(2, 12, 38, 7, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

With period length t, k(j)=residue+t*j is strictly increasing on all nonnegative j, and every selected value is the same period entry. The displayed j only evaluates this infinite index law. The exact result, retaining the requested order, is (2, 12, 38, 7, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0037/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-subsequence · candidate





## 38. A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence

Use the whole infinite real periodic sequence with nonempty period_values=[13, -14, 15/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=0 and subsequence parameter j=13≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate).

View answer and explanation(0, 13, 39, 13, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

With period length t, k(j)=residue+t*j is strictly increasing on all nonnegative j, and every selected value is the same period entry. The displayed j only evaluates this infinite index law. The exact result, retaining the requested order, is (0, 13, 39, 13, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0038/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-subsequence · candidate





## 39. A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence

Use the whole infinite real periodic sequence with nonempty period_values=[14, -15, 16/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=1 and subsequence parameter j=14≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate).

View answer and explanation(1, 14, 43, -15, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

With period length t, k(j)=residue+t*j is strictly increasing on all nonnegative j, and every selected value is the same period entry. The displayed j only evaluates this infinite index law. The exact result, retaining the requested order, is (1, 14, 43, -15, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0039/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-subsequence · candidate





## 40. A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence

Use the whole infinite real periodic sequence with nonempty period_values=[15, -16, 17/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=2 and subsequence parameter j=15≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate).

View answer and explanation(2, 15, 47, 17/2, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

With period length t, k(j)=residue+t*j is strictly increasing on all nonnegative j, and every selected value is the same period entry. The displayed j only evaluates this infinite index law. The exact result, retaining the requested order, is (2, 15, 47, 17/2, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0040/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-subsequence · candidate





## 41. A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence

Use the whole infinite real periodic sequence with nonempty period_values=[16, -17, 18/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=0 and subsequence parameter j=16≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate).

View answer and explanation(0, 16, 48, 16, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

With period length t, k(j)=residue+t*j is strictly increasing on all nonnegative j, and every selected value is the same period entry. The displayed j only evaluates this infinite index law. The exact result, retaining the requested order, is (0, 16, 48, 16, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0041/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-subsequence · candidate





## 42. A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence

Use the whole infinite real periodic sequence with nonempty period_values=[17, -18, 19/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=1 and subsequence parameter j=17≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate).

View answer and explanation(1, 17, 52, -18, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

With period length t, k(j)=residue+t*j is strictly increasing on all nonnegative j, and every selected value is the same period entry. The displayed j only evaluates this infinite index law. The exact result, retaining the requested order, is (1, 17, 52, -18, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0042/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-subsequence · candidate





## 43. A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence

Use the whole infinite real periodic sequence with nonempty period_values=[18, -19, 20/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=2 and subsequence parameter j=18≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate).

View answer and explanation(2, 18, 56, 10, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

With period length t, k(j)=residue+t*j is strictly increasing on all nonnegative j, and every selected value is the same period entry. The displayed j only evaluates this infinite index law. The exact result, retaining the requested order, is (2, 18, 56, 10, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0043/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-subsequence · candidate





## 44. A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence

Use the whole infinite real periodic sequence with nonempty period_values=[19, -20, 21/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=0 and subsequence parameter j=19≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate).

View answer and explanation(0, 19, 57, 19, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

With period length t, k(j)=residue+t*j is strictly increasing on all nonnegative j, and every selected value is the same period entry. The displayed j only evaluates this infinite index law. The exact result, retaining the requested order, is (0, 19, 57, 19, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0044/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-subsequence · candidate





## 45. A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence

Use the whole infinite real periodic sequence with nonempty period_values=[20, -21, 22/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=1 and subsequence parameter j=20≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate).

View answer and explanation(1, 20, 61, -21, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

With period length t, k(j)=residue+t*j is strictly increasing on all nonnegative j, and every selected value is the same period entry. The displayed j only evaluates this infinite index law. The exact result, retaining the requested order, is (1, 20, 61, -21, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0045/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-subsequence · candidate





## 46. A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence

Use the whole infinite real periodic sequence with nonempty period_values=[21, -22, 23/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=2 and subsequence parameter j=21≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate).

View answer and explanation(2, 21, 65, 23/2, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

With period length t, k(j)=residue+t*j is strictly increasing on all nonnegative j, and every selected value is the same period entry. The displayed j only evaluates this infinite index law. The exact result, retaining the requested order, is (2, 21, 65, 23/2, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0046/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-subsequence · candidate





## 47. A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence

Use the whole infinite real periodic sequence with nonempty period_values=[22, -23, 24/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=0 and subsequence parameter j=22≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate).

View answer and explanation(0, 22, 66, 22, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

With period length t, k(j)=residue+t*j is strictly increasing on all nonnegative j, and every selected value is the same period entry. The displayed j only evaluates this infinite index law. The exact result, retaining the requested order, is (0, 22, 66, 22, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0047/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-subsequence · candidate





## 48. A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence

Use the whole infinite real periodic sequence with nonempty period_values=[23, -24, 25/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=1 and subsequence parameter j=23≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate).

View answer and explanation(1, 23, 70, -24, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

With period length t, k(j)=residue+t*j is strictly increasing on all nonnegative j, and every selected value is the same period entry. The displayed j only evaluates this infinite index law. The exact result, retaining the requested order, is (1, 23, 70, -24, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0048/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-subsequence · candidate





## 49. A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence

Use the whole infinite real periodic sequence with nonempty period_values=[24, -25, 26/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=2 and subsequence parameter j=24≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate).

View answer and explanation(2, 24, 74, 13, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

With period length t, k(j)=residue+t*j is strictly increasing on all nonnegative j, and every selected value is the same period entry. The displayed j only evaluates this infinite index law. The exact result, retaining the requested order, is (2, 24, 74, 13, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0049/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-subsequence · candidate





## 50. A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence

Use the whole infinite real periodic sequence with nonempty period_values=[25, -26, 27/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=0 and subsequence parameter j=25≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate).

View answer and explanation(0, 25, 75, 25, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)

With period length t, k(j)=residue+t*j is strictly increasing on all nonnegative j, and every selected value is the same period entry. The displayed j only evaluates this infinite index law. The exact result, retaining the requested order, is (0, 25, 75, 25, Yes, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0050/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-periodic-subsequence · candidate





## 51. Parity Subsequences Need a Strict All-Later Epsilon Witness

Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=1/2, c=1>0, shift=2≥1, parity=0 and epsilon=1/10>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).

View answer and explanation(0, k(j)=2j+0, 1/2, 1/10, 5, 7/12, Yes, forall_j_ge_J_symbolic_subsequence_tail)

Find the least nonnegative J such that c/(2J+parity+shift)<epsilon. Equality still fails the strict inequality. Report a at k(J), not a_J; increasing denominators establish the same bound for every j≥J. Both parity subsequences converge to L. The exact result, retaining the requested order, is (0, k(j)=2j+0, 1/2, 1/10, 5, 7/12, Yes, forall_j_ge_J_symbolic_subsequence_tail).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0051/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-oscillatory-tail · candidate





## 52. Parity Subsequences Need a Strict All-Later Epsilon Witness

Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=2/3, c=2>0, shift=3≥1, parity=1 and epsilon=1/11>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).

View answer and explanation(1, k(j)=2j+1, 2/3, 1/11, 10, 7/12, Yes, forall_j_ge_J_symbolic_subsequence_tail)

Find the least nonnegative J such that c/(2J+parity+shift)<epsilon. Equality still fails the strict inequality. Report a at k(J), not a_J; increasing denominators establish the same bound for every j≥J. Both parity subsequences converge to L. The exact result, retaining the requested order, is (1, k(j)=2j+1, 2/3, 1/11, 10, 7/12, Yes, forall_j_ge_J_symbolic_subsequence_tail).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0052/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-oscillatory-tail · candidate





## 53. Parity Subsequences Need a Strict All-Later Epsilon Witness

Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=3/4, c=3>0, shift=4≥1, parity=0 and epsilon=1/12>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).

View answer and explanation(0, k(j)=2j+0, 3/4, 1/12, 17, 63/76, Yes, forall_j_ge_J_symbolic_subsequence_tail)

Find the least nonnegative J such that c/(2J+parity+shift)<epsilon. Equality still fails the strict inequality. Report a at k(J), not a_J; increasing denominators establish the same bound for every j≥J. Both parity subsequences converge to L. The exact result, retaining the requested order, is (0, k(j)=2j+0, 3/4, 1/12, 17, 63/76, Yes, forall_j_ge_J_symbolic_subsequence_tail).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0053/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-oscillatory-tail · candidate





## 54. Parity Subsequences Need a Strict All-Later Epsilon Witness

Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=4/5, c=4>0, shift=5≥1, parity=1 and epsilon=1/13>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).

View answer and explanation(1, k(j)=2j+1, 4/5, 1/13, 24, 98/135, Yes, forall_j_ge_J_symbolic_subsequence_tail)

Find the least nonnegative J such that c/(2J+parity+shift)<epsilon. Equality still fails the strict inequality. Report a at k(J), not a_J; increasing denominators establish the same bound for every j≥J. Both parity subsequences converge to L. The exact result, retaining the requested order, is (1, k(j)=2j+1, 4/5, 1/13, 24, 98/135, Yes, forall_j_ge_J_symbolic_subsequence_tail).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0054/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-oscillatory-tail · candidate





## 55. Parity Subsequences Need a Strict All-Later Epsilon Witness

Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=5/6, c=5>0, shift=6≥1, parity=0 and epsilon=1/14>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).

View answer and explanation(0, k(j)=2j+0, 5/6, 1/14, 33, 65/72, Yes, forall_j_ge_J_symbolic_subsequence_tail)

Find the least nonnegative J such that c/(2J+parity+shift)<epsilon. Equality still fails the strict inequality. Report a at k(J), not a_J; increasing denominators establish the same bound for every j≥J. Both parity subsequences converge to L. The exact result, retaining the requested order, is (0, k(j)=2j+0, 5/6, 1/14, 33, 65/72, Yes, forall_j_ge_J_symbolic_subsequence_tail).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0055/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-oscillatory-tail · candidate





## 56. Parity Subsequences Need a Strict All-Later Epsilon Witness

Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=6/7, c=6>0, shift=7≥1, parity=1 and epsilon=1/15>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).

View answer and explanation(1, k(j)=2j+1, 6/7, 1/15, 42, 255/322, Yes, forall_j_ge_J_symbolic_subsequence_tail)

Find the least nonnegative J such that c/(2J+parity+shift)<epsilon. Equality still fails the strict inequality. Report a at k(J), not a_J; increasing denominators establish the same bound for every j≥J. Both parity subsequences converge to L. The exact result, retaining the requested order, is (1, k(j)=2j+1, 6/7, 1/15, 42, 255/322, Yes, forall_j_ge_J_symbolic_subsequence_tail).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0056/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-oscillatory-tail · candidate





## 57. Parity Subsequences Need a Strict All-Later Epsilon Witness

Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=7/8, c=7>0, shift=8≥1, parity=0 and epsilon=1/16>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).

View answer and explanation(0, k(j)=2j+0, 7/8, 1/16, 53, 427/456, Yes, forall_j_ge_J_symbolic_subsequence_tail)

Find the least nonnegative J such that c/(2J+parity+shift)<epsilon. Equality still fails the strict inequality. Report a at k(J), not a_J; increasing denominators establish the same bound for every j≥J. Both parity subsequences converge to L. The exact result, retaining the requested order, is (0, k(j)=2j+0, 7/8, 1/16, 53, 427/456, Yes, forall_j_ge_J_symbolic_subsequence_tail).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0057/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-oscillatory-tail · candidate





## 58. Parity Subsequences Need a Strict All-Later Epsilon Witness

Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=8/9, c=8>0, shift=9≥1, parity=1 and epsilon=1/17>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).

View answer and explanation(1, k(j)=2j+1, 8/9, 1/17, 64, 172/207, Yes, forall_j_ge_J_symbolic_subsequence_tail)

Find the least nonnegative J such that c/(2J+parity+shift)<epsilon. Equality still fails the strict inequality. Report a at k(J), not a_J; increasing denominators establish the same bound for every j≥J. Both parity subsequences converge to L. The exact result, retaining the requested order, is (1, k(j)=2j+1, 8/9, 1/17, 64, 172/207, Yes, forall_j_ge_J_symbolic_subsequence_tail).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0058/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-oscillatory-tail · candidate





## 59. Parity Subsequences Need a Strict All-Later Epsilon Witness

Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=9/10, c=9>0, shift=10≥1, parity=0 and epsilon=1/18>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).

View answer and explanation(0, k(j)=2j+0, 9/10, 1/18, 77, 783/820, Yes, forall_j_ge_J_symbolic_subsequence_tail)

Find the least nonnegative J such that c/(2J+parity+shift)<epsilon. Equality still fails the strict inequality. Report a at k(J), not a_J; increasing denominators establish the same bound for every j≥J. Both parity subsequences converge to L. The exact result, retaining the requested order, is (0, k(j)=2j+0, 9/10, 1/18, 77, 783/820, Yes, forall_j_ge_J_symbolic_subsequence_tail).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0059/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-oscillatory-tail · candidate





## 60. Parity Subsequences Need a Strict All-Later Epsilon Witness

Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=10/11, c=10>0, shift=11≥1, parity=1 and epsilon=1/19>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).

View answer and explanation(1, k(j)=2j+1, 10/11, 1/19, 90, 905/1056, Yes, forall_j_ge_J_symbolic_subsequence_tail)

Find the least nonnegative J such that c/(2J+parity+shift)<epsilon. Equality still fails the strict inequality. Report a at k(J), not a_J; increasing denominators establish the same bound for every j≥J. Both parity subsequences converge to L. The exact result, retaining the requested order, is (1, k(j)=2j+1, 10/11, 1/19, 90, 905/1056, Yes, forall_j_ge_J_symbolic_subsequence_tail).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0060/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-oscillatory-tail · candidate





## 61. Parity Subsequences Need a Strict All-Later Epsilon Witness

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).

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

Find the least nonnegative J such that c/(2J+parity+shift)<epsilon. Equality still fails the strict inequality. Report a at k(J), not a_J; increasing denominators establish the same bound for every j≥J. Both parity subsequences converge to L. The exact result, retaining the requested order, is (0, k(j)=2j+0, 11/12, 1/20, 105, 143/148, Yes, forall_j_ge_J_symbolic_subsequence_tail).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0061/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-oscillatory-tail · candidate





## 62. Parity Subsequences Need a Strict All-Later Epsilon Witness

Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=12/13, c=12>0, shift=13≥1, parity=1 and epsilon=1/21>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).

View answer and explanation(1, k(j)=2j+1, 12/13, 1/21, 120, 1446/1651, Yes, forall_j_ge_J_symbolic_subsequence_tail)

Find the least nonnegative J such that c/(2J+parity+shift)<epsilon. Equality still fails the strict inequality. Report a at k(J), not a_J; increasing denominators establish the same bound for every j≥J. Both parity subsequences converge to L. The exact result, retaining the requested order, is (1, k(j)=2j+1, 12/13, 1/21, 120, 1446/1651, Yes, forall_j_ge_J_symbolic_subsequence_tail).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0062/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-oscillatory-tail · candidate





## 63. Parity Subsequences Need a Strict All-Later Epsilon Witness

Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=13/14, c=13>0, shift=14≥1, parity=0 and epsilon=1/22>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).

View answer and explanation(0, k(j)=2j+0, 13/14, 1/22, 137, 1963/2016, Yes, forall_j_ge_J_symbolic_subsequence_tail)

Find the least nonnegative J such that c/(2J+parity+shift)<epsilon. Equality still fails the strict inequality. Report a at k(J), not a_J; increasing denominators establish the same bound for every j≥J. Both parity subsequences converge to L. The exact result, retaining the requested order, is (0, k(j)=2j+0, 13/14, 1/22, 137, 1963/2016, Yes, forall_j_ge_J_symbolic_subsequence_tail).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0063/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-oscillatory-tail · candidate





## 64. Parity Subsequences Need a Strict All-Later Epsilon Witness

Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=14/15, c=14>0, shift=15≥1, parity=1 and epsilon=1/23>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).

View answer and explanation(1, k(j)=2j+1, 14/15, 1/23, 154, 721/810, Yes, forall_j_ge_J_symbolic_subsequence_tail)

Find the least nonnegative J such that c/(2J+parity+shift)<epsilon. Equality still fails the strict inequality. Report a at k(J), not a_J; increasing denominators establish the same bound for every j≥J. Both parity subsequences converge to L. The exact result, retaining the requested order, is (1, k(j)=2j+1, 14/15, 1/23, 154, 721/810, Yes, forall_j_ge_J_symbolic_subsequence_tail).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0064/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-oscillatory-tail · candidate





## 65. Parity Subsequences Need a Strict All-Later Epsilon Witness

Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=15/16, c=15>0, shift=16≥1, parity=0 and epsilon=1/24>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).

View answer and explanation(0, k(j)=2j+0, 15/16, 1/24, 173, 2835/2896, Yes, forall_j_ge_J_symbolic_subsequence_tail)

Find the least nonnegative J such that c/(2J+parity+shift)<epsilon. Equality still fails the strict inequality. Report a at k(J), not a_J; increasing denominators establish the same bound for every j≥J. Both parity subsequences converge to L. The exact result, retaining the requested order, is (0, k(j)=2j+0, 15/16, 1/24, 173, 2835/2896, Yes, forall_j_ge_J_symbolic_subsequence_tail).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0065/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-oscillatory-tail · candidate





## 66. Parity Subsequences Need a Strict All-Later Epsilon Witness

Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=16/17, c=16>0, shift=17≥1, parity=1 and epsilon=1/25>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).

View answer and explanation(1, k(j)=2j+1, 16/17, 1/25, 192, 3080/3417, Yes, forall_j_ge_J_symbolic_subsequence_tail)

Find the least nonnegative J such that c/(2J+parity+shift)<epsilon. Equality still fails the strict inequality. Report a at k(J), not a_J; increasing denominators establish the same bound for every j≥J. Both parity subsequences converge to L. The exact result, retaining the requested order, is (1, k(j)=2j+1, 16/17, 1/25, 192, 3080/3417, Yes, forall_j_ge_J_symbolic_subsequence_tail).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0066/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-oscillatory-tail · candidate





## 67. Parity Subsequences Need a Strict All-Later Epsilon Witness

Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=17/18, c=17>0, shift=18≥1, parity=0 and epsilon=1/26>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).

View answer and explanation(0, k(j)=2j+0, 17/18, 1/26, 213, 1309/1332, Yes, forall_j_ge_J_symbolic_subsequence_tail)

Find the least nonnegative J such that c/(2J+parity+shift)<epsilon. Equality still fails the strict inequality. Report a at k(J), not a_J; increasing denominators establish the same bound for every j≥J. Both parity subsequences converge to L. The exact result, retaining the requested order, is (0, k(j)=2j+0, 17/18, 1/26, 213, 1309/1332, Yes, forall_j_ge_J_symbolic_subsequence_tail).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0067/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-oscillatory-tail · candidate





## 68. Parity Subsequences Need a Strict All-Later Epsilon Witness

Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=18/19, c=18>0, shift=19≥1, parity=1 and epsilon=1/27>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).

View answer and explanation(1, k(j)=2j+1, 18/19, 1/27, 234, 4221/4636, Yes, forall_j_ge_J_symbolic_subsequence_tail)

Find the least nonnegative J such that c/(2J+parity+shift)<epsilon. Equality still fails the strict inequality. Report a at k(J), not a_J; increasing denominators establish the same bound for every j≥J. Both parity subsequences converge to L. The exact result, retaining the requested order, is (1, k(j)=2j+1, 18/19, 1/27, 234, 4221/4636, Yes, forall_j_ge_J_symbolic_subsequence_tail).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0068/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-oscillatory-tail · candidate





## 69. Parity Subsequences Need a Strict All-Later Epsilon Witness

Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=19/20, c=19>0, shift=20≥1, parity=0 and epsilon=1/28>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).

View answer and explanation(0, k(j)=2j+0, 19/20, 1/28, 257, 5263/5340, Yes, forall_j_ge_J_symbolic_subsequence_tail)

Find the least nonnegative J such that c/(2J+parity+shift)<epsilon. Equality still fails the strict inequality. Report a at k(J), not a_J; increasing denominators establish the same bound for every j≥J. Both parity subsequences converge to L. The exact result, retaining the requested order, is (0, k(j)=2j+0, 19/20, 1/28, 257, 5263/5340, Yes, forall_j_ge_J_symbolic_subsequence_tail).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0069/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-oscillatory-tail · candidate





## 70. Parity Subsequences Need a Strict All-Later Epsilon Witness

Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=20/21, c=20>0, shift=21≥1, parity=1 and epsilon=1/29>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).

View answer and explanation(1, k(j)=2j+1, 20/21, 1/29, 280, 1870/2037, Yes, forall_j_ge_J_symbolic_subsequence_tail)

Find the least nonnegative J such that c/(2J+parity+shift)<epsilon. Equality still fails the strict inequality. Report a at k(J), not a_J; increasing denominators establish the same bound for every j≥J. Both parity subsequences converge to L. The exact result, retaining the requested order, is (1, k(j)=2j+1, 20/21, 1/29, 280, 1870/2037, Yes, forall_j_ge_J_symbolic_subsequence_tail).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0070/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-oscillatory-tail · candidate





## 71. Parity Subsequences Need a Strict All-Later Epsilon Witness

Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=21/22, c=21>0, shift=22≥1, parity=0 and epsilon=1/30>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).

View answer and explanation(0, k(j)=2j+0, 21/22, 1/30, 305, 6867/6952, Yes, forall_j_ge_J_symbolic_subsequence_tail)

Find the least nonnegative J such that c/(2J+parity+shift)<epsilon. Equality still fails the strict inequality. Report a at k(J), not a_J; increasing denominators establish the same bound for every j≥J. Both parity subsequences converge to L. The exact result, retaining the requested order, is (0, k(j)=2j+0, 21/22, 1/30, 305, 6867/6952, Yes, forall_j_ge_J_symbolic_subsequence_tail).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0071/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-oscillatory-tail · candidate





## 72. Parity Subsequences Need a Strict All-Later Epsilon Witness

Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=22/23, c=22>0, shift=23≥1, parity=1 and epsilon=1/31>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).

View answer and explanation(1, k(j)=2j+1, 22/23, 1/31, 330, 7271/7866, Yes, forall_j_ge_J_symbolic_subsequence_tail)

Find the least nonnegative J such that c/(2J+parity+shift)<epsilon. Equality still fails the strict inequality. Report a at k(J), not a_J; increasing denominators establish the same bound for every j≥J. Both parity subsequences converge to L. The exact result, retaining the requested order, is (1, k(j)=2j+1, 22/23, 1/31, 330, 7271/7866, Yes, forall_j_ge_J_symbolic_subsequence_tail).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0072/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-oscillatory-tail · candidate





## 73. Parity Subsequences Need a Strict All-Later Epsilon Witness

Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=23/24, c=23>0, shift=24≥1, parity=0 and epsilon=1/32>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).

View answer and explanation(0, k(j)=2j+0, 23/24, 1/32, 357, 2921/2952, Yes, forall_j_ge_J_symbolic_subsequence_tail)

Find the least nonnegative J such that c/(2J+parity+shift)<epsilon. Equality still fails the strict inequality. Report a at k(J), not a_J; increasing denominators establish the same bound for every j≥J. Both parity subsequences converge to L. The exact result, retaining the requested order, is (0, k(j)=2j+0, 23/24, 1/32, 357, 2921/2952, Yes, forall_j_ge_J_symbolic_subsequence_tail).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0073/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-oscillatory-tail · candidate





## 74. Parity Subsequences Need a Strict All-Later Epsilon Witness

Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=24/25, c=24>0, shift=25≥1, parity=1 and epsilon=1/33>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).

View answer and explanation(1, k(j)=2j+1, 24/25, 1/33, 384, 9228/9925, Yes, forall_j_ge_J_symbolic_subsequence_tail)

Find the least nonnegative J such that c/(2J+parity+shift)<epsilon. Equality still fails the strict inequality. Report a at k(J), not a_J; increasing denominators establish the same bound for every j≥J. Both parity subsequences converge to L. The exact result, retaining the requested order, is (1, k(j)=2j+1, 24/25, 1/33, 384, 9228/9925, Yes, forall_j_ge_J_symbolic_subsequence_tail).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0074/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-oscillatory-tail · candidate





## 75. Parity Subsequences Need a Strict All-Later Epsilon Witness

Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=25/26, c=25>0, shift=26≥1, parity=0 and epsilon=1/34>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).

View answer and explanation(0, k(j)=2j+0, 25/26, 1/34, 413, 10975/11076, Yes, forall_j_ge_J_symbolic_subsequence_tail)

Find the least nonnegative J such that c/(2J+parity+shift)<epsilon. Equality still fails the strict inequality. Report a at k(J), not a_J; increasing denominators establish the same bound for every j≥J. Both parity subsequences converge to L. The exact result, retaining the requested order, is (0, k(j)=2j+0, 25/26, 1/34, 413, 10975/11076, Yes, forall_j_ge_J_symbolic_subsequence_tail).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0075/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-oscillatory-tail · candidate





## 76. A Finite Prefix Supplies No Whole-Sequence or Infinite-Subsequence Certificate

Only the finite prefix values=[0, 1, 0, 2] is given; no infinite continuation is specified. Give (prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary).

The three No flags withhold whole-sequence boundedness, infinite-subsequence and convergence certificates from a finite prefix; they do not classify an unspecified infinite continuation.

View answer and explanation([0, 1, 0, 2], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

The three No flags withhold whole-sequence boundedness, an infinite subsequence and convergence certificates. They do not say that every continuation is unbounded or divergent. Bolzano-Weierstrass requires a whole bounded real sequence, an infinite strictly increasing index map and a convergent subsequence; it does not require the original sequence to converge or its subsequential limit to be unique. The exact result, retaining the requested order, is ([0, 1, 0, 2], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0076/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-finite-prefix-boundary · candidate





## 77. A Finite Prefix Supplies No Whole-Sequence or Infinite-Subsequence Certificate

Only the finite prefix values=[1, 2, 1, 3] is given; no infinite continuation is specified. Give (prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary).

The three No flags withhold whole-sequence boundedness, infinite-subsequence and convergence certificates from a finite prefix; they do not classify an unspecified infinite continuation.

View answer and explanation([1, 2, 1, 3], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

The three No flags withhold whole-sequence boundedness, an infinite subsequence and convergence certificates. They do not say that every continuation is unbounded or divergent. Bolzano-Weierstrass requires a whole bounded real sequence, an infinite strictly increasing index map and a convergent subsequence; it does not require the original sequence to converge or its subsequential limit to be unique. The exact result, retaining the requested order, is ([1, 2, 1, 3], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0077/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-finite-prefix-boundary · candidate





## 78. A Finite Prefix Supplies No Whole-Sequence or Infinite-Subsequence Certificate

Only the finite prefix values=[2, 3, 2, 4] is given; no infinite continuation is specified. Give (prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary).

The three No flags withhold whole-sequence boundedness, infinite-subsequence and convergence certificates from a finite prefix; they do not classify an unspecified infinite continuation.

View answer and explanation([2, 3, 2, 4], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

The three No flags withhold whole-sequence boundedness, an infinite subsequence and convergence certificates. They do not say that every continuation is unbounded or divergent. Bolzano-Weierstrass requires a whole bounded real sequence, an infinite strictly increasing index map and a convergent subsequence; it does not require the original sequence to converge or its subsequential limit to be unique. The exact result, retaining the requested order, is ([2, 3, 2, 4], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0078/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-finite-prefix-boundary · candidate





## 79. A Finite Prefix Supplies No Whole-Sequence or Infinite-Subsequence Certificate

Only the finite prefix values=[3, 4, 3, 5] is given; no infinite continuation is specified. Give (prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary).

The three No flags withhold whole-sequence boundedness, infinite-subsequence and convergence certificates from a finite prefix; they do not classify an unspecified infinite continuation.

View answer and explanation([3, 4, 3, 5], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

The three No flags withhold whole-sequence boundedness, an infinite subsequence and convergence certificates. They do not say that every continuation is unbounded or divergent. Bolzano-Weierstrass requires a whole bounded real sequence, an infinite strictly increasing index map and a convergent subsequence; it does not require the original sequence to converge or its subsequential limit to be unique. The exact result, retaining the requested order, is ([3, 4, 3, 5], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0079/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-finite-prefix-boundary · candidate





## 80. A Finite Prefix Supplies No Whole-Sequence or Infinite-Subsequence Certificate

Only the finite prefix values=[4, 5, 4, 6] is given; no infinite continuation is specified. Give (prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary).

The three No flags withhold whole-sequence boundedness, infinite-subsequence and convergence certificates from a finite prefix; they do not classify an unspecified infinite continuation.

View answer and explanation([4, 5, 4, 6], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

The three No flags withhold whole-sequence boundedness, an infinite subsequence and convergence certificates. They do not say that every continuation is unbounded or divergent. Bolzano-Weierstrass requires a whole bounded real sequence, an infinite strictly increasing index map and a convergent subsequence; it does not require the original sequence to converge or its subsequential limit to be unique. The exact result, retaining the requested order, is ([4, 5, 4, 6], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0080/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-finite-prefix-boundary · candidate





## 81. A Finite Prefix Supplies No Whole-Sequence or Infinite-Subsequence Certificate

Only the finite prefix values=[5, 6, 5, 7] is given; no infinite continuation is specified. Give (prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary).

The three No flags withhold whole-sequence boundedness, infinite-subsequence and convergence certificates from a finite prefix; they do not classify an unspecified infinite continuation.

View answer and explanation([5, 6, 5, 7], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

The three No flags withhold whole-sequence boundedness, an infinite subsequence and convergence certificates. They do not say that every continuation is unbounded or divergent. Bolzano-Weierstrass requires a whole bounded real sequence, an infinite strictly increasing index map and a convergent subsequence; it does not require the original sequence to converge or its subsequential limit to be unique. The exact result, retaining the requested order, is ([5, 6, 5, 7], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0081/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-finite-prefix-boundary · candidate





## 82. A Finite Prefix Supplies No Whole-Sequence or Infinite-Subsequence Certificate

Only the finite prefix values=[6, 7, 6, 8] is given; no infinite continuation is specified. Give (prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary).

The three No flags withhold whole-sequence boundedness, infinite-subsequence and convergence certificates from a finite prefix; they do not classify an unspecified infinite continuation.

View answer and explanation([6, 7, 6, 8], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

The three No flags withhold whole-sequence boundedness, an infinite subsequence and convergence certificates. They do not say that every continuation is unbounded or divergent. Bolzano-Weierstrass requires a whole bounded real sequence, an infinite strictly increasing index map and a convergent subsequence; it does not require the original sequence to converge or its subsequential limit to be unique. The exact result, retaining the requested order, is ([6, 7, 6, 8], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0082/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-finite-prefix-boundary · candidate





## 83. A Finite Prefix Supplies No Whole-Sequence or Infinite-Subsequence Certificate

Only the finite prefix values=[7, 8, 7, 9] is given; no infinite continuation is specified. Give (prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary).

The three No flags withhold whole-sequence boundedness, infinite-subsequence and convergence certificates from a finite prefix; they do not classify an unspecified infinite continuation.

View answer and explanation([7, 8, 7, 9], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

The three No flags withhold whole-sequence boundedness, an infinite subsequence and convergence certificates. They do not say that every continuation is unbounded or divergent. Bolzano-Weierstrass requires a whole bounded real sequence, an infinite strictly increasing index map and a convergent subsequence; it does not require the original sequence to converge or its subsequential limit to be unique. The exact result, retaining the requested order, is ([7, 8, 7, 9], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0083/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-finite-prefix-boundary · candidate





## 84. A Finite Prefix Supplies No Whole-Sequence or Infinite-Subsequence Certificate

Only the finite prefix values=[8, 9, 8, 10] is given; no infinite continuation is specified. Give (prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary).

The three No flags withhold whole-sequence boundedness, infinite-subsequence and convergence certificates from a finite prefix; they do not classify an unspecified infinite continuation.

View answer and explanation([8, 9, 8, 10], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

The three No flags withhold whole-sequence boundedness, an infinite subsequence and convergence certificates. They do not say that every continuation is unbounded or divergent. Bolzano-Weierstrass requires a whole bounded real sequence, an infinite strictly increasing index map and a convergent subsequence; it does not require the original sequence to converge or its subsequential limit to be unique. The exact result, retaining the requested order, is ([8, 9, 8, 10], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0084/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-finite-prefix-boundary · candidate





## 85. A Finite Prefix Supplies No Whole-Sequence or Infinite-Subsequence Certificate

Only the finite prefix values=[9, 10, 9, 11] is given; no infinite continuation is specified. Give (prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary).

The three No flags withhold whole-sequence boundedness, infinite-subsequence and convergence certificates from a finite prefix; they do not classify an unspecified infinite continuation.

View answer and explanation([9, 10, 9, 11], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

The three No flags withhold whole-sequence boundedness, an infinite subsequence and convergence certificates. They do not say that every continuation is unbounded or divergent. Bolzano-Weierstrass requires a whole bounded real sequence, an infinite strictly increasing index map and a convergent subsequence; it does not require the original sequence to converge or its subsequential limit to be unique. The exact result, retaining the requested order, is ([9, 10, 9, 11], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0085/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-finite-prefix-boundary · candidate





## 86. A Finite Prefix Supplies No Whole-Sequence or Infinite-Subsequence Certificate

Only the finite prefix values=[10, 11, 10, 12] is given; no infinite continuation is specified. Give (prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary).

The three No flags withhold whole-sequence boundedness, infinite-subsequence and convergence certificates from a finite prefix; they do not classify an unspecified infinite continuation.

View answer and explanation([10, 11, 10, 12], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

The three No flags withhold whole-sequence boundedness, an infinite subsequence and convergence certificates. They do not say that every continuation is unbounded or divergent. Bolzano-Weierstrass requires a whole bounded real sequence, an infinite strictly increasing index map and a convergent subsequence; it does not require the original sequence to converge or its subsequential limit to be unique. The exact result, retaining the requested order, is ([10, 11, 10, 12], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0086/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-finite-prefix-boundary · candidate





## 87. A Finite Prefix Supplies No Whole-Sequence or Infinite-Subsequence Certificate

Only the finite prefix values=[11, 12, 11, 13] is given; no infinite continuation is specified. Give (prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary).

The three No flags withhold whole-sequence boundedness, infinite-subsequence and convergence certificates from a finite prefix; they do not classify an unspecified infinite continuation.

View answer and explanation([11, 12, 11, 13], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

The three No flags withhold whole-sequence boundedness, an infinite subsequence and convergence certificates. They do not say that every continuation is unbounded or divergent. Bolzano-Weierstrass requires a whole bounded real sequence, an infinite strictly increasing index map and a convergent subsequence; it does not require the original sequence to converge or its subsequential limit to be unique. The exact result, retaining the requested order, is ([11, 12, 11, 13], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0087/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-finite-prefix-boundary · candidate





## 88. A Finite Prefix Supplies No Whole-Sequence or Infinite-Subsequence Certificate

Only the finite prefix values=[12, 13, 12, 14] is given; no infinite continuation is specified. Give (prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary).

The three No flags withhold whole-sequence boundedness, infinite-subsequence and convergence certificates from a finite prefix; they do not classify an unspecified infinite continuation.

View answer and explanation([12, 13, 12, 14], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

The three No flags withhold whole-sequence boundedness, an infinite subsequence and convergence certificates. They do not say that every continuation is unbounded or divergent. Bolzano-Weierstrass requires a whole bounded real sequence, an infinite strictly increasing index map and a convergent subsequence; it does not require the original sequence to converge or its subsequential limit to be unique. The exact result, retaining the requested order, is ([12, 13, 12, 14], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0088/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-finite-prefix-boundary · candidate





## 89. A Finite Prefix Supplies No Whole-Sequence or Infinite-Subsequence Certificate

Only the finite prefix values=[13, 14, 13, 15] is given; no infinite continuation is specified. Give (prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary).

The three No flags withhold whole-sequence boundedness, infinite-subsequence and convergence certificates from a finite prefix; they do not classify an unspecified infinite continuation.

View answer and explanation([13, 14, 13, 15], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

The three No flags withhold whole-sequence boundedness, an infinite subsequence and convergence certificates. They do not say that every continuation is unbounded or divergent. Bolzano-Weierstrass requires a whole bounded real sequence, an infinite strictly increasing index map and a convergent subsequence; it does not require the original sequence to converge or its subsequential limit to be unique. The exact result, retaining the requested order, is ([13, 14, 13, 15], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0089/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-finite-prefix-boundary · candidate





## 90. A Finite Prefix Supplies No Whole-Sequence or Infinite-Subsequence Certificate

Only the finite prefix values=[14, 15, 14, 16] is given; no infinite continuation is specified. Give (prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary).

The three No flags withhold whole-sequence boundedness, infinite-subsequence and convergence certificates from a finite prefix; they do not classify an unspecified infinite continuation.

View answer and explanation([14, 15, 14, 16], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

The three No flags withhold whole-sequence boundedness, an infinite subsequence and convergence certificates. They do not say that every continuation is unbounded or divergent. Bolzano-Weierstrass requires a whole bounded real sequence, an infinite strictly increasing index map and a convergent subsequence; it does not require the original sequence to converge or its subsequential limit to be unique. The exact result, retaining the requested order, is ([14, 15, 14, 16], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0090/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-finite-prefix-boundary · candidate





## 91. A Finite Prefix Supplies No Whole-Sequence or Infinite-Subsequence Certificate

Only the finite prefix values=[15, 16, 15, 17] is given; no infinite continuation is specified. Give (prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary).

The three No flags withhold whole-sequence boundedness, infinite-subsequence and convergence certificates from a finite prefix; they do not classify an unspecified infinite continuation.

View answer and explanation([15, 16, 15, 17], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

The three No flags withhold whole-sequence boundedness, an infinite subsequence and convergence certificates. They do not say that every continuation is unbounded or divergent. Bolzano-Weierstrass requires a whole bounded real sequence, an infinite strictly increasing index map and a convergent subsequence; it does not require the original sequence to converge or its subsequential limit to be unique. The exact result, retaining the requested order, is ([15, 16, 15, 17], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0091/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-finite-prefix-boundary · candidate





## 92. A Finite Prefix Supplies No Whole-Sequence or Infinite-Subsequence Certificate

Only the finite prefix values=[16, 17, 16, 18] is given; no infinite continuation is specified. Give (prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary).

The three No flags withhold whole-sequence boundedness, infinite-subsequence and convergence certificates from a finite prefix; they do not classify an unspecified infinite continuation.

View answer and explanation([16, 17, 16, 18], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

The three No flags withhold whole-sequence boundedness, an infinite subsequence and convergence certificates. They do not say that every continuation is unbounded or divergent. Bolzano-Weierstrass requires a whole bounded real sequence, an infinite strictly increasing index map and a convergent subsequence; it does not require the original sequence to converge or its subsequential limit to be unique. The exact result, retaining the requested order, is ([16, 17, 16, 18], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0092/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-finite-prefix-boundary · candidate





## 93. A Finite Prefix Supplies No Whole-Sequence or Infinite-Subsequence Certificate

Only the finite prefix values=[17, 18, 17, 19] is given; no infinite continuation is specified. Give (prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary).

The three No flags withhold whole-sequence boundedness, infinite-subsequence and convergence certificates from a finite prefix; they do not classify an unspecified infinite continuation.

View answer and explanation([17, 18, 17, 19], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

The three No flags withhold whole-sequence boundedness, an infinite subsequence and convergence certificates. They do not say that every continuation is unbounded or divergent. Bolzano-Weierstrass requires a whole bounded real sequence, an infinite strictly increasing index map and a convergent subsequence; it does not require the original sequence to converge or its subsequential limit to be unique. The exact result, retaining the requested order, is ([17, 18, 17, 19], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0093/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-finite-prefix-boundary · candidate





## 94. A Finite Prefix Supplies No Whole-Sequence or Infinite-Subsequence Certificate

Only the finite prefix values=[18, 19, 18, 20] is given; no infinite continuation is specified. Give (prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary).

The three No flags withhold whole-sequence boundedness, infinite-subsequence and convergence certificates from a finite prefix; they do not classify an unspecified infinite continuation.

View answer and explanation([18, 19, 18, 20], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

The three No flags withhold whole-sequence boundedness, an infinite subsequence and convergence certificates. They do not say that every continuation is unbounded or divergent. Bolzano-Weierstrass requires a whole bounded real sequence, an infinite strictly increasing index map and a convergent subsequence; it does not require the original sequence to converge or its subsequential limit to be unique. The exact result, retaining the requested order, is ([18, 19, 18, 20], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0094/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-finite-prefix-boundary · candidate





## 95. A Finite Prefix Supplies No Whole-Sequence or Infinite-Subsequence Certificate

Only the finite prefix values=[19, 20, 19, 21] is given; no infinite continuation is specified. Give (prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary).

The three No flags withhold whole-sequence boundedness, infinite-subsequence and convergence certificates from a finite prefix; they do not classify an unspecified infinite continuation.

View answer and explanation([19, 20, 19, 21], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

The three No flags withhold whole-sequence boundedness, an infinite subsequence and convergence certificates. They do not say that every continuation is unbounded or divergent. Bolzano-Weierstrass requires a whole bounded real sequence, an infinite strictly increasing index map and a convergent subsequence; it does not require the original sequence to converge or its subsequential limit to be unique. The exact result, retaining the requested order, is ([19, 20, 19, 21], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0095/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-finite-prefix-boundary · candidate





## 96. A Finite Prefix Supplies No Whole-Sequence or Infinite-Subsequence Certificate

Only the finite prefix values=[20, 21, 20, 22] is given; no infinite continuation is specified. Give (prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary).

The three No flags withhold whole-sequence boundedness, infinite-subsequence and convergence certificates from a finite prefix; they do not classify an unspecified infinite continuation.

View answer and explanation([20, 21, 20, 22], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

The three No flags withhold whole-sequence boundedness, an infinite subsequence and convergence certificates. They do not say that every continuation is unbounded or divergent. Bolzano-Weierstrass requires a whole bounded real sequence, an infinite strictly increasing index map and a convergent subsequence; it does not require the original sequence to converge or its subsequential limit to be unique. The exact result, retaining the requested order, is ([20, 21, 20, 22], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0096/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-finite-prefix-boundary · candidate





## 97. A Finite Prefix Supplies No Whole-Sequence or Infinite-Subsequence Certificate

Only the finite prefix values=[21, 22, 21, 23] is given; no infinite continuation is specified. Give (prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary).

The three No flags withhold whole-sequence boundedness, infinite-subsequence and convergence certificates from a finite prefix; they do not classify an unspecified infinite continuation.

View answer and explanation([21, 22, 21, 23], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

The three No flags withhold whole-sequence boundedness, an infinite subsequence and convergence certificates. They do not say that every continuation is unbounded or divergent. Bolzano-Weierstrass requires a whole bounded real sequence, an infinite strictly increasing index map and a convergent subsequence; it does not require the original sequence to converge or its subsequential limit to be unique. The exact result, retaining the requested order, is ([21, 22, 21, 23], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0097/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-finite-prefix-boundary · candidate





## 98. A Finite Prefix Supplies No Whole-Sequence or Infinite-Subsequence Certificate

Only the finite prefix values=[22, 23, 22, 24] is given; no infinite continuation is specified. Give (prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary).

The three No flags withhold whole-sequence boundedness, infinite-subsequence and convergence certificates from a finite prefix; they do not classify an unspecified infinite continuation.

View answer and explanation([22, 23, 22, 24], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

The three No flags withhold whole-sequence boundedness, an infinite subsequence and convergence certificates. They do not say that every continuation is unbounded or divergent. Bolzano-Weierstrass requires a whole bounded real sequence, an infinite strictly increasing index map and a convergent subsequence; it does not require the original sequence to converge or its subsequential limit to be unique. The exact result, retaining the requested order, is ([22, 23, 22, 24], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0098/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-finite-prefix-boundary · candidate





## 99. A Finite Prefix Supplies No Whole-Sequence or Infinite-Subsequence Certificate

Only the finite prefix values=[23, 24, 23, 25] is given; no infinite continuation is specified. Give (prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary).

The three No flags withhold whole-sequence boundedness, infinite-subsequence and convergence certificates from a finite prefix; they do not classify an unspecified infinite continuation.

View answer and explanation([23, 24, 23, 25], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

The three No flags withhold whole-sequence boundedness, an infinite subsequence and convergence certificates. They do not say that every continuation is unbounded or divergent. Bolzano-Weierstrass requires a whole bounded real sequence, an infinite strictly increasing index map and a convergent subsequence; it does not require the original sequence to converge or its subsequential limit to be unique. The exact result, retaining the requested order, is ([23, 24, 23, 25], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0099/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-finite-prefix-boundary · candidate





## 100. A Finite Prefix Supplies No Whole-Sequence or Infinite-Subsequence Certificate

Only the finite prefix values=[24, 25, 24, 26] is given; no infinite continuation is specified. Give (prefix, whole_sequence_boundedness_certified, infinite_subsequence_certified, convergence_certified, boundary).

The three No flags withhold whole-sequence boundedness, infinite-subsequence and convergence certificates from a finite prefix; they do not classify an unspecified infinite continuation.

View answer and explanation([24, 25, 24, 26], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)

The three No flags withhold whole-sequence boundedness, an infinite subsequence and convergence certificates. They do not say that every continuation is unbounded or divergent. Bolzano-Weierstrass requires a whole bounded real sequence, an infinite strictly increasing index map and a convergent subsequence; it does not require the original sequence to converge or its subsequential limit to be unique. The exact result, retaining the requested order, is ([24, 25, 24, 26], No, No, No, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0100/)

Bolzano-Weierstrass theorem for real sequences · qf-mod40-bw-finite-prefix-boundary · candidate



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