# Question 13467: Distinguish a Completeness Theorem from Sampled Cauchy Sequences

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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.

R0025-13467 · q-20261014-advance-20261028-r03-real-banach-space-0092



## Original question and givens

Use the explicit real coordinate space R^n of dimension=17, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=sampled_cauchy_sequences. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.


- G001 — Use the explicit real coordinate space R^n of dimension=17, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. evidence_mode=sampled_cauchy_sequences. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.





## 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-real-banach-space](https://ocme.evemisslab.com/en/mko/mko-real-banach-space/)





## Step-by-step derivation

S001 · impliesBefore

named real space R^17;norm=l1;evidence=sampled_cauchy_sequences

After

sufficient evidence=No;sample insufficiency is not incompleteness

Conditions: none

The named universal theorem and coordinate argument cover every Cauchy sequence;finite samples do not.No in sample mode neither declares this known space incomplete nor supplies a formal-verification receipt.

Step identifiers and dependenciesRule: Banach_evidence_quantifier_scope · Uses: G001









## Reverse checks


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





## Result

(17, l1, sampled_cauchy_sequences, No, samples_insufficient_for_completeness)

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: 0367b5302a79cafa0bcbfc3f1d5c8a6dfb79325c0815b0d572ee0cab838a8827
- candidate record SHA-256: 343760aab6d3517141d59ebbc426de66637d28ede11601fbeb4f0fd51b388e2b
- semantic review receipt SHA-256: 0e51c2bf621e6e998415fdd75d260464e625cecd5894a230c9f09553249551c5
- question-first witness SHA-256: ceb79401a62782c9f7e5ed86afee9b1a71455f799bd24cf485cba7aa8ecaca8a
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261028-r03-real-banach-space-0092.json)