# Symbolic Deviation Tails, Finite-Prefix Limits and Eventual Equality

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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-20261014-advance-20261030-r07-convergence-in-probability-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Evaluate the Deviation Profile Only at a Valid Probability Index

Use the declared rational deviation-probability tail p_n=1/(n+2) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=1. Give (numerator,shift,n,deviation_probability).

View answer and explanation(1, 2, 1, 1/3)

The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (1, 2, 1, 1/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0001/)

Convergence in probability · qf-mod30-convprob-deviation-probability · candidate





## 2. Evaluate the Deviation Profile Only at a Valid Probability Index

Use the declared rational deviation-probability tail p_n=2/(n+3) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=2. Give (numerator,shift,n,deviation_probability).

View answer and explanation(2, 3, 2, 2/5)

The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (2, 3, 2, 2/5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0002/)

Convergence in probability · qf-mod30-convprob-deviation-probability · candidate





## 3. Evaluate the Deviation Profile Only at a Valid Probability Index

Use the declared rational deviation-probability tail p_n=3/(n+4) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=3. Give (numerator,shift,n,deviation_probability).

View answer and explanation(3, 4, 3, 3/7)

The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (3, 4, 3, 3/7).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0003/)

Convergence in probability · qf-mod30-convprob-deviation-probability · candidate





## 4. Evaluate the Deviation Profile Only at a Valid Probability Index

Use the declared rational deviation-probability tail p_n=4/(n+5) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=4. Give (numerator,shift,n,deviation_probability).

View answer and explanation(4, 5, 4, 4/9)

The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (4, 5, 4, 4/9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0004/)

Convergence in probability · qf-mod30-convprob-deviation-probability · candidate





## 5. Evaluate the Deviation Profile Only at a Valid Probability Index

Use the declared rational deviation-probability tail p_n=5/(n+6) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=5. Give (numerator,shift,n,deviation_probability).

View answer and explanation(5, 6, 5, 5/11)

The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (5, 6, 5, 5/11).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0005/)

Convergence in probability · qf-mod30-convprob-deviation-probability · candidate





## 6. Evaluate the Deviation Profile Only at a Valid Probability Index

Use the declared rational deviation-probability tail p_n=1/(n+7) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=6. Give (numerator,shift,n,deviation_probability).

View answer and explanation(1, 7, 6, 1/13)

The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (1, 7, 6, 1/13).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0006/)

Convergence in probability · qf-mod30-convprob-deviation-probability · candidate





## 7. Evaluate the Deviation Profile Only at a Valid Probability Index

Use the declared rational deviation-probability tail p_n=2/(n+8) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=7. Give (numerator,shift,n,deviation_probability).

View answer and explanation(2, 8, 7, 2/15)

The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (2, 8, 7, 2/15).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0007/)

Convergence in probability · qf-mod30-convprob-deviation-probability · candidate





## 8. Evaluate the Deviation Profile Only at a Valid Probability Index

Use the declared rational deviation-probability tail p_n=3/(n+2) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=8. Give (numerator,shift,n,deviation_probability).

View answer and explanation(3, 2, 8, 3/10)

The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (3, 2, 8, 3/10).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0008/)

Convergence in probability · qf-mod30-convprob-deviation-probability · candidate





## 9. Evaluate the Deviation Profile Only at a Valid Probability Index

Use the declared rational deviation-probability tail p_n=4/(n+3) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=9. Give (numerator,shift,n,deviation_probability).

View answer and explanation(4, 3, 9, 1/3)

The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (4, 3, 9, 1/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0009/)

Convergence in probability · qf-mod30-convprob-deviation-probability · candidate





## 10. Evaluate the Deviation Profile Only at a Valid Probability Index

Use the declared rational deviation-probability tail p_n=5/(n+4) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=10. Give (numerator,shift,n,deviation_probability).

View answer and explanation(5, 4, 10, 5/14)

The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (5, 4, 10, 5/14).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0010/)

Convergence in probability · qf-mod30-convprob-deviation-probability · candidate





## 11. Evaluate the Deviation Profile Only at a Valid Probability Index

Use the declared rational deviation-probability tail p_n=1/(n+5) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=11. Give (numerator,shift,n,deviation_probability).

View answer and explanation(1, 5, 11, 1/16)

The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (1, 5, 11, 1/16).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0011/)

Convergence in probability · qf-mod30-convprob-deviation-probability · candidate





## 12. Evaluate the Deviation Profile Only at a Valid Probability Index

Use the declared rational deviation-probability tail p_n=2/(n+6) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=12. Give (numerator,shift,n,deviation_probability).

View answer and explanation(2, 6, 12, 1/9)

The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (2, 6, 12, 1/9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0012/)

Convergence in probability · qf-mod30-convprob-deviation-probability · candidate





## 13. Evaluate the Deviation Profile Only at a Valid Probability Index

Use the declared rational deviation-probability tail p_n=3/(n+7) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=13. Give (numerator,shift,n,deviation_probability).

View answer and explanation(3, 7, 13, 3/20)

The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (3, 7, 13, 3/20).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0013/)

Convergence in probability · qf-mod30-convprob-deviation-probability · candidate





## 14. Evaluate the Deviation Profile Only at a Valid Probability Index

Use the declared rational deviation-probability tail p_n=4/(n+8) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=14. Give (numerator,shift,n,deviation_probability).

View answer and explanation(4, 8, 14, 2/11)

The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (4, 8, 14, 2/11).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0014/)

Convergence in probability · qf-mod30-convprob-deviation-probability · candidate





## 15. Evaluate the Deviation Profile Only at a Valid Probability Index

Use the declared rational deviation-probability tail p_n=5/(n+2) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=15. Give (numerator,shift,n,deviation_probability).

View answer and explanation(5, 2, 15, 5/17)

The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (5, 2, 15, 5/17).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0015/)

Convergence in probability · qf-mod30-convprob-deviation-probability · candidate





## 16. Evaluate the Deviation Profile Only at a Valid Probability Index

Use the declared rational deviation-probability tail p_n=1/(n+3) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=16. Give (numerator,shift,n,deviation_probability).

View answer and explanation(1, 3, 16, 1/19)

The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (1, 3, 16, 1/19).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0016/)

Convergence in probability · qf-mod30-convprob-deviation-probability · candidate





## 17. Evaluate the Deviation Profile Only at a Valid Probability Index

Use the declared rational deviation-probability tail p_n=2/(n+4) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=17. Give (numerator,shift,n,deviation_probability).

View answer and explanation(2, 4, 17, 2/21)

The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (2, 4, 17, 2/21).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0017/)

Convergence in probability · qf-mod30-convprob-deviation-probability · candidate





## 18. Evaluate the Deviation Profile Only at a Valid Probability Index

Use the declared rational deviation-probability tail p_n=3/(n+5) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=18. Give (numerator,shift,n,deviation_probability).

View answer and explanation(3, 5, 18, 3/23)

The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (3, 5, 18, 3/23).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0018/)

Convergence in probability · qf-mod30-convprob-deviation-probability · candidate





## 19. Evaluate the Deviation Profile Only at a Valid Probability Index

Use the declared rational deviation-probability tail p_n=4/(n+6) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=19. Give (numerator,shift,n,deviation_probability).

View answer and explanation(4, 6, 19, 4/25)

The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (4, 6, 19, 4/25).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0019/)

Convergence in probability · qf-mod30-convprob-deviation-probability · candidate





## 20. Evaluate the Deviation Profile Only at a Valid Probability Index

Use the declared rational deviation-probability tail p_n=5/(n+7) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=20. Give (numerator,shift,n,deviation_probability).

View answer and explanation(5, 7, 20, 5/27)

The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (5, 7, 20, 5/27).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0020/)

Convergence in probability · qf-mod30-convprob-deviation-probability · candidate





## 21. Evaluate the Deviation Profile Only at a Valid Probability Index

Use the declared rational deviation-probability tail p_n=1/(n+8) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=21. Give (numerator,shift,n,deviation_probability).

View answer and explanation(1, 8, 21, 1/29)

The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (1, 8, 21, 1/29).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0021/)

Convergence in probability · qf-mod30-convprob-deviation-probability · candidate





## 22. Evaluate the Deviation Profile Only at a Valid Probability Index

Use the declared rational deviation-probability tail p_n=2/(n+2) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=22. Give (numerator,shift,n,deviation_probability).

View answer and explanation(2, 2, 22, 1/12)

The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (2, 2, 22, 1/12).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0022/)

Convergence in probability · qf-mod30-convprob-deviation-probability · candidate





## 23. Evaluate the Deviation Profile Only at a Valid Probability Index

Use the declared rational deviation-probability tail p_n=3/(n+3) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=23. Give (numerator,shift,n,deviation_probability).

View answer and explanation(3, 3, 23, 3/26)

The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (3, 3, 23, 3/26).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0023/)

Convergence in probability · qf-mod30-convprob-deviation-probability · candidate





## 24. Evaluate the Deviation Profile Only at a Valid Probability Index

Use the declared rational deviation-probability tail p_n=4/(n+4) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=24. Give (numerator,shift,n,deviation_probability).

View answer and explanation(4, 4, 24, 1/7)

The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (4, 4, 24, 1/7).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0024/)

Convergence in probability · qf-mod30-convprob-deviation-probability · candidate





## 25. Evaluate the Deviation Profile Only at a Valid Probability Index

Use the declared rational deviation-probability tail p_n=5/(n+5) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=25. Give (numerator,shift,n,deviation_probability).

View answer and explanation(5, 5, 25, 1/6)

The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (5, 5, 25, 1/6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0025/)

Convergence in probability · qf-mod30-convprob-deviation-probability · candidate





## 26. A Strict Rational Tail Cutoff Is Different from a Finite Prefix

Use the declared rational deviation-probability tail p_n=1/(n+2) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/10. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).

View answer and explanation(1, 2, 1/10, 9, 1/11, Yes, forall_n_ge_N_tail_bound)

Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (1, 2, 1/10, 9, 1/11, Yes, forall_n_ge_N_tail_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0026/)

Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate





## 27. A Strict Rational Tail Cutoff Is Different from a Finite Prefix

Use the declared rational deviation-probability tail p_n=2/(n+3) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/11. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).

View answer and explanation(2, 3, 1/11, 20, 2/23, Yes, forall_n_ge_N_tail_bound)

Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (2, 3, 1/11, 20, 2/23, Yes, forall_n_ge_N_tail_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0027/)

Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate





## 28. A Strict Rational Tail Cutoff Is Different from a Finite Prefix

Use the declared rational deviation-probability tail p_n=3/(n+4) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/12. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).

View answer and explanation(3, 4, 1/12, 33, 3/37, Yes, forall_n_ge_N_tail_bound)

Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (3, 4, 1/12, 33, 3/37, Yes, forall_n_ge_N_tail_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0028/)

Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate





## 29. A Strict Rational Tail Cutoff Is Different from a Finite Prefix

Use the declared rational deviation-probability tail p_n=4/(n+5) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/13. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).

View answer and explanation(4, 5, 1/13, 48, 4/53, Yes, forall_n_ge_N_tail_bound)

Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (4, 5, 1/13, 48, 4/53, Yes, forall_n_ge_N_tail_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0029/)

Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate





## 30. A Strict Rational Tail Cutoff Is Different from a Finite Prefix

Use the declared rational deviation-probability tail p_n=5/(n+6) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/14. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).

View answer and explanation(5, 6, 1/14, 65, 5/71, Yes, forall_n_ge_N_tail_bound)

Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (5, 6, 1/14, 65, 5/71, Yes, forall_n_ge_N_tail_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0030/)

Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate





## 31. A Strict Rational Tail Cutoff Is Different from a Finite Prefix

Use the declared rational deviation-probability tail p_n=1/(n+7) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/15. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).

View answer and explanation(1, 7, 1/15, 9, 1/16, Yes, forall_n_ge_N_tail_bound)

Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (1, 7, 1/15, 9, 1/16, Yes, forall_n_ge_N_tail_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0031/)

Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate





## 32. A Strict Rational Tail Cutoff Is Different from a Finite Prefix

Use the declared rational deviation-probability tail p_n=2/(n+8) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/16. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).

View answer and explanation(2, 8, 1/16, 25, 2/33, Yes, forall_n_ge_N_tail_bound)

Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (2, 8, 1/16, 25, 2/33, Yes, forall_n_ge_N_tail_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0032/)

Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate





## 33. A Strict Rational Tail Cutoff Is Different from a Finite Prefix

Use the declared rational deviation-probability tail p_n=3/(n+2) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/17. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).

View answer and explanation(3, 2, 1/17, 50, 3/52, Yes, forall_n_ge_N_tail_bound)

Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (3, 2, 1/17, 50, 3/52, Yes, forall_n_ge_N_tail_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0033/)

Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate





## 34. A Strict Rational Tail Cutoff Is Different from a Finite Prefix

Use the declared rational deviation-probability tail p_n=4/(n+3) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/18. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).

View answer and explanation(4, 3, 1/18, 70, 4/73, Yes, forall_n_ge_N_tail_bound)

Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (4, 3, 1/18, 70, 4/73, Yes, forall_n_ge_N_tail_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0034/)

Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate





## 35. A Strict Rational Tail Cutoff Is Different from a Finite Prefix

Use the declared rational deviation-probability tail p_n=5/(n+4) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/19. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).

View answer and explanation(5, 4, 1/19, 92, 5/96, Yes, forall_n_ge_N_tail_bound)

Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (5, 4, 1/19, 92, 5/96, Yes, forall_n_ge_N_tail_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0035/)

Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate





## 36. A Strict Rational Tail Cutoff Is Different from a Finite Prefix

Use the declared rational deviation-probability tail p_n=1/(n+5) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/20. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).

View answer and explanation(1, 5, 1/20, 16, 1/21, Yes, forall_n_ge_N_tail_bound)

Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (1, 5, 1/20, 16, 1/21, Yes, forall_n_ge_N_tail_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0036/)

Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate





## 37. A Strict Rational Tail Cutoff Is Different from a Finite Prefix

Use the declared rational deviation-probability tail p_n=2/(n+6) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/21. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).

View answer and explanation(2, 6, 1/21, 37, 2/43, Yes, forall_n_ge_N_tail_bound)

Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (2, 6, 1/21, 37, 2/43, Yes, forall_n_ge_N_tail_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0037/)

Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate





## 38. A Strict Rational Tail Cutoff Is Different from a Finite Prefix

Use the declared rational deviation-probability tail p_n=3/(n+7) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/22. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).

View answer and explanation(3, 7, 1/22, 60, 3/67, Yes, forall_n_ge_N_tail_bound)

Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (3, 7, 1/22, 60, 3/67, Yes, forall_n_ge_N_tail_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0038/)

Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate





## 39. A Strict Rational Tail Cutoff Is Different from a Finite Prefix

Use the declared rational deviation-probability tail p_n=4/(n+8) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/23. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).

View answer and explanation(4, 8, 1/23, 85, 4/93, Yes, forall_n_ge_N_tail_bound)

Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (4, 8, 1/23, 85, 4/93, Yes, forall_n_ge_N_tail_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0039/)

Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate





## 40. A Strict Rational Tail Cutoff Is Different from a Finite Prefix

Use the declared rational deviation-probability tail p_n=5/(n+2) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/24. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).

View answer and explanation(5, 2, 1/24, 119, 5/121, Yes, forall_n_ge_N_tail_bound)

Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (5, 2, 1/24, 119, 5/121, Yes, forall_n_ge_N_tail_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0040/)

Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate





## 41. A Strict Rational Tail Cutoff Is Different from a Finite Prefix

Use the declared rational deviation-probability tail p_n=1/(n+3) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/25. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).

View answer and explanation(1, 3, 1/25, 23, 1/26, Yes, forall_n_ge_N_tail_bound)

Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (1, 3, 1/25, 23, 1/26, Yes, forall_n_ge_N_tail_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0041/)

Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate





## 42. A Strict Rational Tail Cutoff Is Different from a Finite Prefix

Use the declared rational deviation-probability tail p_n=2/(n+4) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/26. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).

View answer and explanation(2, 4, 1/26, 49, 2/53, Yes, forall_n_ge_N_tail_bound)

Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (2, 4, 1/26, 49, 2/53, Yes, forall_n_ge_N_tail_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0042/)

Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate





## 43. A Strict Rational Tail Cutoff Is Different from a Finite Prefix

Use the declared rational deviation-probability tail p_n=3/(n+5) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/27. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).

View answer and explanation(3, 5, 1/27, 77, 3/82, Yes, forall_n_ge_N_tail_bound)

Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (3, 5, 1/27, 77, 3/82, Yes, forall_n_ge_N_tail_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0043/)

Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate





## 44. A Strict Rational Tail Cutoff Is Different from a Finite Prefix

Use the declared rational deviation-probability tail p_n=4/(n+6) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/28. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).

View answer and explanation(4, 6, 1/28, 107, 4/113, Yes, forall_n_ge_N_tail_bound)

Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (4, 6, 1/28, 107, 4/113, Yes, forall_n_ge_N_tail_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0044/)

Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate





## 45. A Strict Rational Tail Cutoff Is Different from a Finite Prefix

Use the declared rational deviation-probability tail p_n=5/(n+7) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/29. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).

View answer and explanation(5, 7, 1/29, 139, 5/146, Yes, forall_n_ge_N_tail_bound)

Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (5, 7, 1/29, 139, 5/146, Yes, forall_n_ge_N_tail_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0045/)

Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate





## 46. A Strict Rational Tail Cutoff Is Different from a Finite Prefix

Use the declared rational deviation-probability tail p_n=1/(n+8) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/30. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).

View answer and explanation(1, 8, 1/30, 23, 1/31, Yes, forall_n_ge_N_tail_bound)

Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (1, 8, 1/30, 23, 1/31, Yes, forall_n_ge_N_tail_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0046/)

Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate





## 47. A Strict Rational Tail Cutoff Is Different from a Finite Prefix

Use the declared rational deviation-probability tail p_n=2/(n+2) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/31. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).

View answer and explanation(2, 2, 1/31, 61, 2/63, Yes, forall_n_ge_N_tail_bound)

Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (2, 2, 1/31, 61, 2/63, Yes, forall_n_ge_N_tail_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0047/)

Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate





## 48. A Strict Rational Tail Cutoff Is Different from a Finite Prefix

Use the declared rational deviation-probability tail p_n=3/(n+3) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/32. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).

View answer and explanation(3, 3, 1/32, 94, 3/97, Yes, forall_n_ge_N_tail_bound)

Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (3, 3, 1/32, 94, 3/97, Yes, forall_n_ge_N_tail_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0048/)

Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate





## 49. A Strict Rational Tail Cutoff Is Different from a Finite Prefix

Use the declared rational deviation-probability tail p_n=4/(n+4) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/33. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).

View answer and explanation(4, 4, 1/33, 129, 4/133, Yes, forall_n_ge_N_tail_bound)

Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (4, 4, 1/33, 129, 4/133, Yes, forall_n_ge_N_tail_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0049/)

Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate





## 50. A Strict Rational Tail Cutoff Is Different from a Finite Prefix

Use the declared rational deviation-probability tail p_n=5/(n+5) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/34. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).

View answer and explanation(5, 5, 1/34, 166, 5/171, Yes, forall_n_ge_N_tail_bound)

Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (5, 5, 1/34, 166, 5/171, Yes, forall_n_ge_N_tail_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0050/)

Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate





## 51. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

Use only the explicitly supplied finite list p_n=1/(n+2) for n=0 through prefix_length-1, with prefix_length=1. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1, 2, 1, [1/2], No, finite_prefix_not_convergence_certificate)

List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (1, 2, 1, [1/2], No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0051/)

Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate





## 52. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

Use only the explicitly supplied finite list p_n=2/(n+3) for n=0 through prefix_length-1, with prefix_length=2. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(2, 3, 2, [2/3, 1/2], No, finite_prefix_not_convergence_certificate)

List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (2, 3, 2, [2/3, 1/2], No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0052/)

Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate





## 53. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

Use only the explicitly supplied finite list p_n=3/(n+4) for n=0 through prefix_length-1, with prefix_length=3. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(3, 4, 3, [3/4, 3/5, 1/2], No, finite_prefix_not_convergence_certificate)

List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (3, 4, 3, [3/4, 3/5, 1/2], No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0053/)

Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate





## 54. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

Use only the explicitly supplied finite list p_n=4/(n+5) for n=0 through prefix_length-1, with prefix_length=4. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(4, 5, 4, [4/5, 2/3, 4/7, 1/2], No, finite_prefix_not_convergence_certificate)

List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (4, 5, 4, [4/5, 2/3, 4/7, 1/2], No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0054/)

Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate





## 55. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

Use only the explicitly supplied finite list p_n=5/(n+6) for n=0 through prefix_length-1, with prefix_length=5. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(5, 6, 5, [5/6, 5/7, 5/8, 5/9, 1/2], No, finite_prefix_not_convergence_certificate)

List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (5, 6, 5, [5/6, 5/7, 5/8, 5/9, 1/2], No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0055/)

Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate





## 56. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

Use only the explicitly supplied finite list p_n=1/(n+7) for n=0 through prefix_length-1, with prefix_length=6. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1, 7, 6, [1/7, 1/8, 1/9, 1/10, 1/11, 1/12], No, finite_prefix_not_convergence_certificate)

List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (1, 7, 6, [1/7, 1/8, 1/9, 1/10, 1/11, 1/12], No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0056/)

Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate





## 57. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

Use only the explicitly supplied finite list p_n=2/(n+8) for n=0 through prefix_length-1, with prefix_length=7. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(2, 8, 7, [1/4, 2/9, 1/5, 2/11, 1/6, 2/13, 1/7], No, finite_prefix_not_convergence_certificate)

List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (2, 8, 7, [1/4, 2/9, 1/5, 2/11, 1/6, 2/13, 1/7], No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0057/)

Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate





## 58. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

Use only the explicitly supplied finite list p_n=3/(n+3) for n=0 through prefix_length-1, with prefix_length=8. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(3, 3, 8, [1, 3/4, 3/5, 1/2, 3/7, 3/8, 1/3, 3/10], No, finite_prefix_not_convergence_certificate)

List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (3, 3, 8, [1, 3/4, 3/5, 1/2, 3/7, 3/8, 1/3, 3/10], No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0058/)

Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate





## 59. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

Use only the explicitly supplied finite list p_n=4/(n+4) for n=0 through prefix_length-1, with prefix_length=9. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(4, 4, 9, [1, 4/5, 2/3, 4/7, 1/2, 4/9, 2/5, 4/11, 1/3], No, finite_prefix_not_convergence_certificate)

List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (4, 4, 9, [1, 4/5, 2/3, 4/7, 1/2, 4/9, 2/5, 4/11, 1/3], No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0059/)

Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate





## 60. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

Use only the explicitly supplied finite list p_n=5/(n+5) for n=0 through prefix_length-1, with prefix_length=10. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(5, 5, 10, [1, 5/6, 5/7, 5/8, 5/9, 1/2, 5/11, 5/12, 5/13, 5/14], No, finite_prefix_not_convergence_certificate)

List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (5, 5, 10, [1, 5/6, 5/7, 5/8, 5/9, 1/2, 5/11, 5/12, 5/13, 5/14], No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0060/)

Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate





## 61. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

Use only the explicitly supplied finite list p_n=1/(n+5) for n=0 through prefix_length-1, with prefix_length=11. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1, 5, 11, [1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15], No, finite_prefix_not_convergence_certificate)

List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (1, 5, 11, [1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15], No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0061/)

Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate





## 62. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

Use only the explicitly supplied finite list p_n=2/(n+6) for n=0 through prefix_length-1, with prefix_length=12. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(2, 6, 12, [1/3, 2/7, 1/4, 2/9, 1/5, 2/11, 1/6, 2/13, 1/7, 2/15, 1/8, 2/17], No, finite_prefix_not_convergence_certificate)

List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (2, 6, 12, [1/3, 2/7, 1/4, 2/9, 1/5, 2/11, 1/6, 2/13, 1/7, 2/15, 1/8, 2/17], No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0062/)

Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate





## 63. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

Use only the explicitly supplied finite list p_n=3/(n+7) for n=0 through prefix_length-1, with prefix_length=13. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(3, 7, 13, [3/7, 3/8, 1/3, 3/10, 3/11, 1/4, 3/13, 3/14, 1/5, 3/16, 3/17, 1/6, 3/19], No, finite_prefix_not_convergence_certificate)

List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (3, 7, 13, [3/7, 3/8, 1/3, 3/10, 3/11, 1/4, 3/13, 3/14, 1/5, 3/16, 3/17, 1/6, 3/19], No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0063/)

Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate





## 64. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

Use only the explicitly supplied finite list p_n=4/(n+8) for n=0 through prefix_length-1, with prefix_length=14. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(4, 8, 14, [1/2, 4/9, 2/5, 4/11, 1/3, 4/13, 2/7, 4/15, 1/4, 4/17, 2/9, 4/19, 1/5, 4/21], No, finite_prefix_not_convergence_certificate)

List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (4, 8, 14, [1/2, 4/9, 2/5, 4/11, 1/3, 4/13, 2/7, 4/15, 1/4, 4/17, 2/9, 4/19, 1/5, 4/21], No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0064/)

Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate





## 65. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

Use only the explicitly supplied finite list p_n=5/(n+5) for n=0 through prefix_length-1, with prefix_length=15. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(5, 5, 15, [1, 5/6, 5/7, 5/8, 5/9, 1/2, 5/11, 5/12, 5/13, 5/14, 1/3, 5/16, 5/17, 5/18, 5/19], No, finite_prefix_not_convergence_certificate)

List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (5, 5, 15, [1, 5/6, 5/7, 5/8, 5/9, 1/2, 5/11, 5/12, 5/13, 5/14, 1/3, 5/16, 5/17, 5/18, 5/19], No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0065/)

Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate





## 66. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

Use only the explicitly supplied finite list p_n=1/(n+3) for n=0 through prefix_length-1, with prefix_length=16. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1, 3, 16, [1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15, 1/16, 1/17, 1/18], No, finite_prefix_not_convergence_certificate)

List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (1, 3, 16, [1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15, 1/16, 1/17, 1/18], No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0066/)

Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate





## 67. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

Use only the explicitly supplied finite list p_n=2/(n+4) for n=0 through prefix_length-1, with prefix_length=17. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(2, 4, 17, [1/2, 2/5, 1/3, 2/7, 1/4, 2/9, 1/5, 2/11, 1/6, 2/13, 1/7, 2/15, 1/8, 2/17, 1/9, 2/19, 1/10], No, finite_prefix_not_convergence_certificate)

List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (2, 4, 17, [1/2, 2/5, 1/3, 2/7, 1/4, 2/9, 1/5, 2/11, 1/6, 2/13, 1/7, 2/15, 1/8, 2/17, 1/9, 2/19, 1/10], No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0067/)

Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate





## 68. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

Use only the explicitly supplied finite list p_n=3/(n+5) for n=0 through prefix_length-1, with prefix_length=18. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(3, 5, 18, [3/5, 1/2, 3/7, 3/8, 1/3, 3/10, 3/11, 1/4, 3/13, 3/14, 1/5, 3/16, 3/17, 1/6, 3/19, 3/20, 1/7, 3/22], No, finite_prefix_not_convergence_certificate)

List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (3, 5, 18, [3/5, 1/2, 3/7, 3/8, 1/3, 3/10, 3/11, 1/4, 3/13, 3/14, 1/5, 3/16, 3/17, 1/6, 3/19, 3/20, 1/7, 3/22], No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0068/)

Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate





## 69. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

Use only the explicitly supplied finite list p_n=4/(n+6) for n=0 through prefix_length-1, with prefix_length=19. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(4, 6, 19, [2/3, 4/7, 1/2, 4/9, 2/5, 4/11, 1/3, 4/13, 2/7, 4/15, 1/4, 4/17, 2/9, 4/19, 1/5, 4/21, 2/11, 4/23, 1/6], No, finite_prefix_not_convergence_certificate)

List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (4, 6, 19, [2/3, 4/7, 1/2, 4/9, 2/5, 4/11, 1/3, 4/13, 2/7, 4/15, 1/4, 4/17, 2/9, 4/19, 1/5, 4/21, 2/11, 4/23, 1/6], No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0069/)

Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate





## 70. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

Use only the explicitly supplied finite list p_n=5/(n+7) for n=0 through prefix_length-1, with prefix_length=20. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(5, 7, 20, [5/7, 5/8, 5/9, 1/2, 5/11, 5/12, 5/13, 5/14, 1/3, 5/16, 5/17, 5/18, 5/19, 1/4, 5/21, 5/22, 5/23, 5/24, 1/5, 5/26], No, finite_prefix_not_convergence_certificate)

List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (5, 7, 20, [5/7, 5/8, 5/9, 1/2, 5/11, 5/12, 5/13, 5/14, 1/3, 5/16, 5/17, 5/18, 5/19, 1/4, 5/21, 5/22, 5/23, 5/24, 1/5, 5/26], No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0070/)

Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate





## 71. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

Use only the explicitly supplied finite list p_n=1/(n+8) for n=0 through prefix_length-1, with prefix_length=21. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1, 8, 21, [1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15, 1/16, 1/17, 1/18, 1/19, 1/20, 1/21, 1/22, 1/23, 1/24, 1/25, 1/26, 1/27, 1/28], No, finite_prefix_not_convergence_certificate)

List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (1, 8, 21, [1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15, 1/16, 1/17, 1/18, 1/19, 1/20, 1/21, 1/22, 1/23, 1/24, 1/25, 1/26, 1/27, 1/28], No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0071/)

Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate





## 72. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

Use only the explicitly supplied finite list p_n=2/(n+2) for n=0 through prefix_length-1, with prefix_length=22. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(2, 2, 22, [1, 2/3, 1/2, 2/5, 1/3, 2/7, 1/4, 2/9, 1/5, 2/11, 1/6, 2/13, 1/7, 2/15, 1/8, 2/17, 1/9, 2/19, 1/10, 2/21, 1/11, 2/23], No, finite_prefix_not_convergence_certificate)

List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (2, 2, 22, [1, 2/3, 1/2, 2/5, 1/3, 2/7, 1/4, 2/9, 1/5, 2/11, 1/6, 2/13, 1/7, 2/15, 1/8, 2/17, 1/9, 2/19, 1/10, 2/21, 1/11, 2/23], No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0072/)

Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate





## 73. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

Use only the explicitly supplied finite list p_n=3/(n+3) for n=0 through prefix_length-1, with prefix_length=23. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(3, 3, 23, [1, 3/4, 3/5, 1/2, 3/7, 3/8, 1/3, 3/10, 3/11, 1/4, 3/13, 3/14, 1/5, 3/16, 3/17, 1/6, 3/19, 3/20, 1/7, 3/22, 3/23, 1/8, 3/25], No, finite_prefix_not_convergence_certificate)

List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (3, 3, 23, [1, 3/4, 3/5, 1/2, 3/7, 3/8, 1/3, 3/10, 3/11, 1/4, 3/13, 3/14, 1/5, 3/16, 3/17, 1/6, 3/19, 3/20, 1/7, 3/22, 3/23, 1/8, 3/25], No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0073/)

Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate





## 74. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

Use only the explicitly supplied finite list p_n=4/(n+4) for n=0 through prefix_length-1, with prefix_length=24. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(4, 4, 24, [1, 4/5, 2/3, 4/7, 1/2, 4/9, 2/5, 4/11, 1/3, 4/13, 2/7, 4/15, 1/4, 4/17, 2/9, 4/19, 1/5, 4/21, 2/11, 4/23, 1/6, 4/25, 2/13, 4/27], No, finite_prefix_not_convergence_certificate)

List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (4, 4, 24, [1, 4/5, 2/3, 4/7, 1/2, 4/9, 2/5, 4/11, 1/3, 4/13, 2/7, 4/15, 1/4, 4/17, 2/9, 4/19, 1/5, 4/21, 2/11, 4/23, 1/6, 4/25, 2/13, 4/27], No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0074/)

Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate





## 75. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate

Use only the explicitly supplied finite list p_n=5/(n+5) for n=0 through prefix_length-1, with prefix_length=25. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(5, 5, 25, [1, 5/6, 5/7, 5/8, 5/9, 1/2, 5/11, 5/12, 5/13, 5/14, 1/3, 5/16, 5/17, 5/18, 5/19, 1/4, 5/21, 5/22, 5/23, 5/24, 1/5, 5/26, 5/27, 5/28, 5/29], No, finite_prefix_not_convergence_certificate)

List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (5, 5, 25, [1, 5/6, 5/7, 5/8, 5/9, 1/2, 5/11, 5/12, 5/13, 5/14, 1/3, 5/16, 5/17, 5/18, 5/19, 1/4, 5/21, 5/22, 5/23, 5/24, 1/5, 5/26, 5/27, 5/28, 5/29], No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0075/)

Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate





## 76. Eventual Equality Supplies a Conditional Universal-Tail Certificate

On the declared common probability space assume X_n=X almost surely for every n≥settling_index=1; inspect n=0. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).

View answer and explanation(1, 0, not_yet_zero, Yes, eventual_equality_universal_tail_certificate)

At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (1, 0, not_yet_zero, Yes, eventual_equality_universal_tail_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0076/)

Convergence in probability · qf-mod30-convprob-eventual-equality · candidate





## 77. Eventual Equality Supplies a Conditional Universal-Tail Certificate

On the declared common probability space assume X_n=X almost surely for every n≥settling_index=2; inspect n=1. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).

View answer and explanation(2, 1, not_yet_zero, Yes, eventual_equality_universal_tail_certificate)

At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (2, 1, not_yet_zero, Yes, eventual_equality_universal_tail_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0077/)

Convergence in probability · qf-mod30-convprob-eventual-equality · candidate





## 78. Eventual Equality Supplies a Conditional Universal-Tail Certificate

On the declared common probability space assume X_n=X almost surely for every n≥settling_index=3; inspect n=2. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).

View answer and explanation(3, 2, not_yet_zero, Yes, eventual_equality_universal_tail_certificate)

At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (3, 2, not_yet_zero, Yes, eventual_equality_universal_tail_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0078/)

Convergence in probability · qf-mod30-convprob-eventual-equality · candidate





## 79. Eventual Equality Supplies a Conditional Universal-Tail Certificate

On the declared common probability space assume X_n=X almost surely for every n≥settling_index=4; inspect n=4. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).

View answer and explanation(4, 4, 0, Yes, eventual_equality_universal_tail_certificate)

At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (4, 4, 0, Yes, eventual_equality_universal_tail_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0079/)

Convergence in probability · qf-mod30-convprob-eventual-equality · candidate





## 80. Eventual Equality Supplies a Conditional Universal-Tail Certificate

On the declared common probability space assume X_n=X almost surely for every n≥settling_index=5; inspect n=5. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).

View answer and explanation(5, 5, 0, Yes, eventual_equality_universal_tail_certificate)

At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (5, 5, 0, Yes, eventual_equality_universal_tail_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0080/)

Convergence in probability · qf-mod30-convprob-eventual-equality · candidate





## 81. Eventual Equality Supplies a Conditional Universal-Tail Certificate

On the declared common probability space assume X_n=X almost surely for every n≥settling_index=6; inspect n=6. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).

View answer and explanation(6, 6, 0, Yes, eventual_equality_universal_tail_certificate)

At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (6, 6, 0, Yes, eventual_equality_universal_tail_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0081/)

Convergence in probability · qf-mod30-convprob-eventual-equality · candidate





## 82. Eventual Equality Supplies a Conditional Universal-Tail Certificate

On the declared common probability space assume X_n=X almost surely for every n≥settling_index=7; inspect n=8. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).

View answer and explanation(7, 8, 0, Yes, eventual_equality_universal_tail_certificate)

At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (7, 8, 0, Yes, eventual_equality_universal_tail_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0082/)

Convergence in probability · qf-mod30-convprob-eventual-equality · candidate





## 83. Eventual Equality Supplies a Conditional Universal-Tail Certificate

On the declared common probability space assume X_n=X almost surely for every n≥settling_index=8; inspect n=9. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).

View answer and explanation(8, 9, 0, Yes, eventual_equality_universal_tail_certificate)

At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (8, 9, 0, Yes, eventual_equality_universal_tail_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0083/)

Convergence in probability · qf-mod30-convprob-eventual-equality · candidate





## 84. Eventual Equality Supplies a Conditional Universal-Tail Certificate

On the declared common probability space assume X_n=X almost surely for every n≥settling_index=9; inspect n=10. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).

View answer and explanation(9, 10, 0, Yes, eventual_equality_universal_tail_certificate)

At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (9, 10, 0, Yes, eventual_equality_universal_tail_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0084/)

Convergence in probability · qf-mod30-convprob-eventual-equality · candidate





## 85. Eventual Equality Supplies a Conditional Universal-Tail Certificate

On the declared common probability space assume X_n=X almost surely for every n≥settling_index=10; inspect n=12. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).

View answer and explanation(10, 12, 0, Yes, eventual_equality_universal_tail_certificate)

At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (10, 12, 0, Yes, eventual_equality_universal_tail_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0085/)

Convergence in probability · qf-mod30-convprob-eventual-equality · candidate





## 86. Eventual Equality Supplies a Conditional Universal-Tail Certificate

On the declared common probability space assume X_n=X almost surely for every n≥settling_index=11; inspect n=13. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).

View answer and explanation(11, 13, 0, Yes, eventual_equality_universal_tail_certificate)

At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (11, 13, 0, Yes, eventual_equality_universal_tail_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0086/)

Convergence in probability · qf-mod30-convprob-eventual-equality · candidate





## 87. Eventual Equality Supplies a Conditional Universal-Tail Certificate

On the declared common probability space assume X_n=X almost surely for every n≥settling_index=12; inspect n=14. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).

View answer and explanation(12, 14, 0, Yes, eventual_equality_universal_tail_certificate)

At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (12, 14, 0, Yes, eventual_equality_universal_tail_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0087/)

Convergence in probability · qf-mod30-convprob-eventual-equality · candidate





## 88. Eventual Equality Supplies a Conditional Universal-Tail Certificate

On the declared common probability space assume X_n=X almost surely for every n≥settling_index=13; inspect n=16. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).

View answer and explanation(13, 16, 0, Yes, eventual_equality_universal_tail_certificate)

At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (13, 16, 0, Yes, eventual_equality_universal_tail_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0088/)

Convergence in probability · qf-mod30-convprob-eventual-equality · candidate





## 89. Eventual Equality Supplies a Conditional Universal-Tail Certificate

On the declared common probability space assume X_n=X almost surely for every n≥settling_index=14; inspect n=17. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).

View answer and explanation(14, 17, 0, Yes, eventual_equality_universal_tail_certificate)

At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (14, 17, 0, Yes, eventual_equality_universal_tail_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0089/)

Convergence in probability · qf-mod30-convprob-eventual-equality · candidate





## 90. Eventual Equality Supplies a Conditional Universal-Tail Certificate

On the declared common probability space assume X_n=X almost surely for every n≥settling_index=15; inspect n=18. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).

View answer and explanation(15, 18, 0, Yes, eventual_equality_universal_tail_certificate)

At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (15, 18, 0, Yes, eventual_equality_universal_tail_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0090/)

Convergence in probability · qf-mod30-convprob-eventual-equality · candidate





## 91. Eventual Equality Supplies a Conditional Universal-Tail Certificate

On the declared common probability space assume X_n=X almost surely for every n≥settling_index=16; inspect n=20. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).

View answer and explanation(16, 20, 0, Yes, eventual_equality_universal_tail_certificate)

At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (16, 20, 0, Yes, eventual_equality_universal_tail_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0091/)

Convergence in probability · qf-mod30-convprob-eventual-equality · candidate





## 92. Eventual Equality Supplies a Conditional Universal-Tail Certificate

On the declared common probability space assume X_n=X almost surely for every n≥settling_index=17; inspect n=21. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).

View answer and explanation(17, 21, 0, Yes, eventual_equality_universal_tail_certificate)

At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (17, 21, 0, Yes, eventual_equality_universal_tail_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0092/)

Convergence in probability · qf-mod30-convprob-eventual-equality · candidate





## 93. Eventual Equality Supplies a Conditional Universal-Tail Certificate

On the declared common probability space assume X_n=X almost surely for every n≥settling_index=18; inspect n=22. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).

View answer and explanation(18, 22, 0, Yes, eventual_equality_universal_tail_certificate)

At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (18, 22, 0, Yes, eventual_equality_universal_tail_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0093/)

Convergence in probability · qf-mod30-convprob-eventual-equality · candidate





## 94. Eventual Equality Supplies a Conditional Universal-Tail Certificate

On the declared common probability space assume X_n=X almost surely for every n≥settling_index=19; inspect n=24. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).

View answer and explanation(19, 24, 0, Yes, eventual_equality_universal_tail_certificate)

At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (19, 24, 0, Yes, eventual_equality_universal_tail_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0094/)

Convergence in probability · qf-mod30-convprob-eventual-equality · candidate





## 95. Eventual Equality Supplies a Conditional Universal-Tail Certificate

On the declared common probability space assume X_n=X almost surely for every n≥settling_index=20; inspect n=25. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).

View answer and explanation(20, 25, 0, Yes, eventual_equality_universal_tail_certificate)

At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (20, 25, 0, Yes, eventual_equality_universal_tail_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0095/)

Convergence in probability · qf-mod30-convprob-eventual-equality · candidate





## 96. Eventual Equality Supplies a Conditional Universal-Tail Certificate

On the declared common probability space assume X_n=X almost surely for every n≥settling_index=21; inspect n=26. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).

View answer and explanation(21, 26, 0, Yes, eventual_equality_universal_tail_certificate)

At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (21, 26, 0, Yes, eventual_equality_universal_tail_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0096/)

Convergence in probability · qf-mod30-convprob-eventual-equality · candidate





## 97. Eventual Equality Supplies a Conditional Universal-Tail Certificate

On the declared common probability space assume X_n=X almost surely for every n≥settling_index=22; inspect n=28. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).

View answer and explanation(22, 28, 0, Yes, eventual_equality_universal_tail_certificate)

At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (22, 28, 0, Yes, eventual_equality_universal_tail_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0097/)

Convergence in probability · qf-mod30-convprob-eventual-equality · candidate





## 98. Eventual Equality Supplies a Conditional Universal-Tail Certificate

On the declared common probability space assume X_n=X almost surely for every n≥settling_index=23; inspect n=29. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).

View answer and explanation(23, 29, 0, Yes, eventual_equality_universal_tail_certificate)

At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (23, 29, 0, Yes, eventual_equality_universal_tail_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0098/)

Convergence in probability · qf-mod30-convprob-eventual-equality · candidate





## 99. Eventual Equality Supplies a Conditional Universal-Tail Certificate

On the declared common probability space assume X_n=X almost surely for every n≥settling_index=24; inspect n=30. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).

View answer and explanation(24, 30, 0, Yes, eventual_equality_universal_tail_certificate)

At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (24, 30, 0, Yes, eventual_equality_universal_tail_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0099/)

Convergence in probability · qf-mod30-convprob-eventual-equality · candidate





## 100. Eventual Equality Supplies a Conditional Universal-Tail Certificate

On the declared common probability space assume X_n=X almost surely for every n≥settling_index=25; inspect n=32. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).

View answer and explanation(25, 32, 0, Yes, eventual_equality_universal_tail_certificate)

At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (25, 32, 0, Yes, eventual_equality_universal_tail_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0100/)

Convergence in probability · qf-mod30-convprob-eventual-equality · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/revisions/ocme-parameter-revision-20261017/batch-20261014-advance-20261030-r07-convergence-in-probability-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261030-r07-convergence-in-probability-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/revisions/ocme-parameter-revision-20261017/batch-20261014-advance-20261030-r07-convergence-in-probability-foundations/validation-report.json)