Question 13451: Distinguish a Completeness Theorem from Sampled Cauchy Sequences

Original question and givens

Use the explicit real coordinate space R^n of dimension=1, 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=1, 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

Step-by-step derivation

named real space R^1;norm=l1;evidence=sampled_cauchy_sequences

sufficient evidence=No;sample insufficiency is not incompleteness

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

(1, 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.