# Question 21472: A Whole Declared Periodic Sequence Has an Exact Absolute Bound

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0041-21472 · q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0022



## Original question and givens

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).


- G001 — 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).





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-bolzano-weierstrass-theorem-real](https://ocme.evemisslab.com/en/mko/mko-bolzano-weierstrass-theorem-real/)





## Step-by-step derivation

S001 · impliesBefore

source operation=bw-periodic-boundedness;original snapshot and mathematics retained

After

explicit ordered source output fields=[period_values, absolute_bound, whole_sequence_bounded, certificate]

Conditions: none

Retain the historical interface marker while listing ordered output fields without JavaScript.Fields and tags are not extra mathematical evidence;fixed No markers retain their claim scope.

Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · impliesBefore

ORIGINAL period index=0;value=22;absolute value=22

After

prefix absolute maximum after this original entry=22

Conditions: none

Scan absolute values over the full period rather than the maximum signed value.A negative entry may determine the bound;retain original positions and duplicates.

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





S003 · impliesBefore

ORIGINAL period index=1;value=-23;absolute value=23

After

prefix absolute maximum after this original entry=23

Conditions: none

Scan absolute values over the full period rather than the maximum signed value.A negative entry may determine the bound;retain original positions and duplicates.

Step identifiers and dependenciesRule: bw_every_period_entry_absolute_bound · Uses: G001, S002





S004 · impliesBefore

ORIGINAL period index=2;value=12;absolute value=12

After

prefix absolute maximum after this original entry=23

Conditions: none

Scan absolute values over the full period rather than the maximum signed value.A negative entry may determine the bound;retain original positions and duplicates.

Step identifiers and dependenciesRule: bw_every_period_entry_absolute_bound · Uses: G001, S003





S005 · impliesBefore

ORIGINAL NONEMPTY full period=[22, -23, 12];period length=3

After

for EVERY n>=0,unique n=3q+r with0<=r<3 gives a_n=period_values[r];therefore |a_n|<=23

Conditions: none

Whole-sequence boundedness follows from the declared infinite periodic law and division with remainder,not observed future behavior.This proves the stated model,not the general Bolzano-Weierstrass theorem from finite samples.

Step identifiers and dependenciesRule: bw_periodic_law_for_all_n_not_finite_observation · Uses: G001, S004









## Reverse checks


- S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

([22, -23, 12], 23, Yes, whole_infinite_periodic_sequence_bounded)

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





## This question review and source binding

Review date: 2026-10-26

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

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: e29220fac11893e54d8bdfe5b7796a5a34e089196a49a98e17bfb5d7c7ebbb82
- candidate record SHA-256: 665cbdaa974544671cdc00bdd65cae232cb4e0f0f3ccd630cfee73efa37afd8b
- semantic review receipt SHA-256: 8f134403487694af4c70d2e00ec84bab542043ef4b3c54517c04f3c409c0262a
- question-first witness SHA-256: ad772845088ed8e99f87f90d4264b05ea97be35775305a02011af5ee35f0f4ce
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0022.json)