# Finite Tail Audits, Infinite Cauchy Witnesses and Alternating Counterexamples This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed. 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-20261013-advance-20261026-r03-cauchy-sequence-foundations · 2026-10-13 · 100 questions This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt. ## 1. A Finite Tail Audit Cannot Establish General Cauchy Behavior Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes) For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-12, 1/2, -3], -5, 3, for any epsilon>0 choose N=prefix length, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0026/) Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate ## 27. An Explicit Infinite Constant Tail Supplies a Cauchy Witness Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes) For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-11, 1, -1], -2, 3, for any epsilon>0 choose N=prefix length, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0027/) Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate ## 28. An Explicit Infinite Constant Tail Supplies a Cauchy Witness Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes) For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-10, 3/2, -1/3], -1, 3, for any epsilon>0 choose N=prefix length, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0028/) Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate ## 29. An Explicit Infinite Constant Tail Supplies a Cauchy Witness Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes) For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-9, 2, 0], -1/2, 3, for any epsilon>0 choose N=prefix length, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0029/) Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate ## 30. An Explicit Infinite Constant Tail Supplies a Cauchy Witness Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes) For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-8, 5/2, 1/5], -1/5, 3, for any epsilon>0 choose N=prefix length, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0030/) Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate ## 31. An Explicit Infinite Constant Tail Supplies a Cauchy Witness Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes) For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-7, 3, 2], 0, 3, for any epsilon>0 choose N=prefix length, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0031/) Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate ## 32. An Explicit Infinite Constant Tail Supplies a Cauchy Witness Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes) For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-6, 7/2, 3/2], 1/7, 3, for any epsilon>0 choose N=prefix length, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0032/) Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate ## 33. An Explicit Infinite Constant Tail Supplies a Cauchy Witness Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes) For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-5, 4, -1], 2, 3, for any epsilon>0 choose N=prefix length, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0033/) Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate ## 34. An Explicit Infinite Constant Tail Supplies a Cauchy Witness Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes) For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-4, 9/2, -1/2], 3/2, 3, for any epsilon>0 choose N=prefix length, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0034/) Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate ## 35. An Explicit Infinite Constant Tail Supplies a Cauchy Witness Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes) For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-3, 5, -1/5], 4/3, 3, for any epsilon>0 choose N=prefix length, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0035/) Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate ## 36. An Explicit Infinite Constant Tail Supplies a Cauchy Witness Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes) For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-2, 11/2, 0], 5/4, 3, for any epsilon>0 choose N=prefix length, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0036/) Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate ## 37. An Explicit Infinite Constant Tail Supplies a Cauchy Witness Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes) For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-1, 6, 1/2], -1, 3, for any epsilon>0 choose N=prefix length, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0037/) Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate ## 38. An Explicit Infinite Constant Tail Supplies a Cauchy Witness Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes) For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 13/2, 2/3], -2/3, 3, for any epsilon>0 choose N=prefix length, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0038/) Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate ## 39. An Explicit Infinite Constant Tail Supplies a Cauchy Witness Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes) For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 7, 3/4], -3/7, 3, for any epsilon>0 choose N=prefix length, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0039/) Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate ## 40. An Explicit Infinite Constant Tail Supplies a Cauchy Witness Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes) For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 15/2, -3/5], -2, 3, for any epsilon>0 choose N=prefix length, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0040/) Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate ## 41. An Explicit Infinite Constant Tail Supplies a Cauchy Witness Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes) For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([3, 8, -2], -1/2, 3, for any epsilon>0 choose N=prefix length, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0041/) Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate ## 42. An Explicit Infinite Constant Tail Supplies a Cauchy Witness Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes) For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([4, 17/2, -1/2], 0, 3, for any epsilon>0 choose N=prefix length, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0042/) Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate ## 43. An Explicit Infinite Constant Tail Supplies a Cauchy Witness Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes) For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([5, 9, 0], 1/4, 3, for any epsilon>0 choose N=prefix length, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0043/) Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate ## 44. An Explicit Infinite Constant Tail Supplies a Cauchy Witness Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes) For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([6, 19/2, 1/4], 2/5, 3, for any epsilon>0 choose N=prefix length, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0044/) Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate ## 45. An Explicit Infinite Constant Tail Supplies a Cauchy Witness Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes) For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([7, 10, 2/5], 1/2, 3, for any epsilon>0 choose N=prefix length, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0045/) Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate ## 46. An Explicit Infinite Constant Tail Supplies a Cauchy Witness Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes) For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([8, 21/2, 3], 4/7, 3, for any epsilon>0 choose N=prefix length, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0046/) Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate ## 47. An Explicit Infinite Constant Tail Supplies a Cauchy Witness Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes) For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([9, 11, -3/2], 5, 3, for any epsilon>0 choose N=prefix length, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0047/) Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate ## 48. An Explicit Infinite Constant Tail Supplies a Cauchy Witness Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes) For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([10, 23/2, -2/3], -5/2, 3, for any epsilon>0 choose N=prefix length, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0048/) Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate ## 49. An Explicit Infinite Constant Tail Supplies a Cauchy Witness Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes) For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([11, 12, -1/4], -4/3, 3, for any epsilon>0 choose N=prefix length, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0049/) Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate ## 50. An Explicit Infinite Constant Tail Supplies a Cauchy Witness Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes) For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([12, 25/2, 0], -3/4, 3, for any epsilon>0 choose N=prefix length, Yes). [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r03-cauchy-sequence-0050/) Cauchy sequence · qf-mod26-cauchy-ultimately-constant-certificate · candidate ## 51. A Strict Reciprocal Bound for the Selected Epsilon Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)