# Question 12804: A Non-Cauchy Sequence Does Not Contradict Space Completeness [Return to catalogue page 65](https://ocme.evemisslab.com/en/questions/page/65/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261027-r03-complete-metric-space-foundations/#q-20261014-advance-20261027-r03-complete-metric-space-0054) 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. R0024-12804 · q-20261014-advance-20261027-r03-complete-metric-space-0054 ## Original question and givens Use the complete qualified finite metric table=[[0, 6, 13, 17], [6, 0, 11, 15], [13, 11, 0, 22], [17, 15, 22, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=3 and right=0, with arbitrary proposed cutoff N=7. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms. - G001 — Use the complete qualified finite metric table=[[0, 6, 13, 17], [6, 0, 11, 15], [13, 11, 0, 22], [17, 15, 22, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=3 and right=0, with arbitrary proposed cutoff N=7. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms. ## 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-complete-metric-space](https://ocme.evemisslab.com/en/mko/mko-complete-metric-space/) ## Step-by-step derivation S001 · impliesBefore carrier Fin(4);complete nonnegative integer metric candidate=[[0, 6, 13, 17], [6, 0, 11, 15], [13, 11, 0, 22], [17, 15, 22, 0]] After zero iff equal=Yes;symmetric=Yes Conditions: none First qualify nonnegativity,diagonal/off-diagonal separation and symmetry on the entire table,not only queried cells.At least two points is a premise of this minimum-positive-distance protocol,not a claim that a singleton is incomplete. Step identifiers and dependenciesRule: complete_metric_separation_and_symmetry · Uses: G001 S002 · impliesBefore endpoints=[0, 0];direct distance=0;all intermediate labels=[0, 1, 2, 3] After all two-leg distances=[0, 12, 26, 34];direct<=every two-leg=Yes Conditions: none For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles. Step identifiers and dependenciesRule: complete_metric_all_triangle_intermediates · Uses: G001, S001 S003 · impliesBefore endpoints=[0, 1];direct distance=6;all intermediate labels=[0, 1, 2, 3] After all two-leg distances=[6, 6, 24, 32];direct<=every two-leg=Yes Conditions: none For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles. Step identifiers and dependenciesRule: complete_metric_all_triangle_intermediates · Uses: G001, S002 S004 · impliesBefore endpoints=[0, 2];direct distance=13;all intermediate labels=[0, 1, 2, 3] After all two-leg distances=[13, 17, 13, 39];direct<=every two-leg=Yes Conditions: none For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles. Step identifiers and dependenciesRule: complete_metric_all_triangle_intermediates · Uses: G001, S003 S005 · impliesBefore endpoints=[0, 3];direct distance=17;all intermediate labels=[0, 1, 2, 3] After all two-leg distances=[17, 21, 35, 17];direct<=every two-leg=Yes Conditions: none For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles. Step identifiers and dependenciesRule: complete_metric_all_triangle_intermediates · Uses: G001, S004 S006 · impliesBefore endpoints=[1, 0];direct distance=6;all intermediate labels=[0, 1, 2, 3] After all two-leg distances=[6, 6, 24, 32];direct<=every two-leg=Yes Conditions: none For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles. Step identifiers and dependenciesRule: complete_metric_all_triangle_intermediates · Uses: G001, S005 S007 · impliesBefore endpoints=[1, 1];direct distance=0;all intermediate labels=[0, 1, 2, 3] After all two-leg distances=[12, 0, 22, 30];direct<=every two-leg=Yes Conditions: none For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles. Step identifiers and dependenciesRule: complete_metric_all_triangle_intermediates · Uses: G001, S006 S008 · impliesBefore endpoints=[1, 2];direct distance=11;all intermediate labels=[0, 1, 2, 3] After all two-leg distances=[19, 11, 11, 37];direct<=every two-leg=Yes Conditions: none For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles. Step identifiers and dependenciesRule: complete_metric_all_triangle_intermediates · Uses: G001, S007 S009 · impliesBefore endpoints=[1, 3];direct distance=15;all intermediate labels=[0, 1, 2, 3] After all two-leg distances=[23, 15, 33, 15];direct<=every two-leg=Yes Conditions: none For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles. Step identifiers and dependenciesRule: complete_metric_all_triangle_intermediates · Uses: G001, S008 S010 · impliesBefore endpoints=[2, 0];direct distance=13;all intermediate labels=[0, 1, 2, 3] After all two-leg distances=[13, 17, 13, 39];direct<=every two-leg=Yes Conditions: none For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles. Step identifiers and dependenciesRule: complete_metric_all_triangle_intermediates · Uses: G001, S009 S011 · impliesBefore endpoints=[2, 1];direct distance=11;all intermediate labels=[0, 1, 2, 3] After all two-leg distances=[19, 11, 11, 37];direct<=every two-leg=Yes Conditions: none For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles. Step identifiers and dependenciesRule: complete_metric_all_triangle_intermediates · Uses: G001, S010 S012 · impliesBefore endpoints=[2, 2];direct distance=0;all intermediate labels=[0, 1, 2, 3] After all two-leg distances=[26, 22, 0, 44];direct<=every two-leg=Yes Conditions: none For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles. Step identifiers and dependenciesRule: complete_metric_all_triangle_intermediates · Uses: G001, S011 S013 · impliesBefore endpoints=[2, 3];direct distance=22;all intermediate labels=[0, 1, 2, 3] After all two-leg distances=[30, 26, 22, 22];direct<=every two-leg=Yes Conditions: none For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles. Step identifiers and dependenciesRule: complete_metric_all_triangle_intermediates · Uses: G001, S012 S014 · impliesBefore endpoints=[3, 0];direct distance=17;all intermediate labels=[0, 1, 2, 3] After all two-leg distances=[17, 21, 35, 17];direct<=every two-leg=Yes Conditions: none For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles. Step identifiers and dependenciesRule: complete_metric_all_triangle_intermediates · Uses: G001, S013 S015 · impliesBefore endpoints=[3, 1];direct distance=15;all intermediate labels=[0, 1, 2, 3] After all two-leg distances=[23, 15, 33, 15];direct<=every two-leg=Yes Conditions: none For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles. Step identifiers and dependenciesRule: complete_metric_all_triangle_intermediates · Uses: G001, S014 S016 · impliesBefore endpoints=[3, 2];direct distance=22;all intermediate labels=[0, 1, 2, 3] After all two-leg distances=[30, 26, 22, 22];direct<=every two-leg=Yes Conditions: none For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles. Step identifiers and dependenciesRule: complete_metric_all_triangle_intermediates · Uses: G001, S015 S017 · impliesBefore endpoints=[3, 3];direct distance=0;all intermediate labels=[0, 1, 2, 3] After all two-leg distances=[34, 30, 44, 0];direct<=every two-leg=Yes Conditions: none For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles. Step identifiers and dependenciesRule: complete_metric_all_triangle_intermediates · Uses: G001, S016 S018 · impliesBefore all off-diagonal positive distances=[6, 13, 17, 6, 11, 15, 13, 11, 22, 17, 15, 22] After minimum positive distance delta=6>0 Conditions: none A finite nonempty list of positive distances has a positive minimum.This is the space-wide scale for the universal Cauchy certificate,not a threshold guessed from a sequence. Step identifiers and dependenciesRule: finite_metric_positive_minimum_separation · Uses: G001, S017 S019 · calculationBefore adjacent alternating tail index := sum(7, 1) After adjacent alternating tail index = 8 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · impliesBefore explicit infinite sequence alternates labels=[3, 0];arbitrary cutoff=7 After indices=[7, 8];epsilon=d(left,right)=17>0;actual adjacent distance=17=epsilon Conditions: none Adjacent terms are these two points in one order,and symmetry gives the same distance.Fix positive epsilon=d.Equality with epsilon violates strict inequality,and this construction works for every cutoff N. Step identifiers and dependenciesRule: metric_alternation_fixed_epsilon_strict_failure · Uses: G001, S019 S021 · impliesBefore every cutoff defeated at fixed epsilon=17 After counterexample=Yes;Cauchy=No;non-Cauchy sequence does not refute space completeness Conditions: none Completeness concerns only Cauchy sequences.A non-Cauchy sequence does not meet that premise.The final field uses the corrected source Cauchy label,not not_cauchy. Step identifiers and dependenciesRule: nonCauchy_not_counterexample_to_completeness · Uses: G001, S020 ## Reverse checks - S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Recompute adjacent alternating tail index with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 (3, 0, 7, 8, 17, 17, Yes, No) 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: b36e62f8eb21e77f30c302f764e6f6038e510f749cd79db90a1cdcc558a941ca - candidate record SHA-256: 9643cff28cfade41c4455e052f56cd4b8770de1c2853a4472a0b89b0fb178b48 - semantic review receipt SHA-256: 1c8f6c2c6c75852cb93c08235684ce9f32ebe86065c58c693a04257e772c2d35 - question-first witness SHA-256: 05fd764b5f9f7fff6185003efe576d87632f6daf40606ac3839a7feafe4f80d8 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261027-r03-complete-metric-space-0054.json)