# Bernoulli CDF Tails, Limit Continuity Points and Finite-Prefix Limits

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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-20261031-r03-convergence-in-distribution-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. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=2, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=1, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation(1, 31/60, -1, 0, 0, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (1, 31/60, -1, 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0001/)

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate





## 2. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=3, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=2, threshold=1/2. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation(2, 13/25, 1/2, 12/25, 1/2, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (2, 13/25, 1/2, 12/25, 1/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0002/)

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate





## 3. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=4, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=3, threshold=2/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation(3, 73/140, 2, 1, 1, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (3, 73/140, 2, 1, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0003/)

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate





## 4. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=5, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=4, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation(4, 47/90, -1, 0, 0, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (4, 47/90, -1, 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0004/)

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate





## 5. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=6, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=5, threshold=1/2. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation(5, 23/44, 1/2, 21/44, 1/2, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (5, 23/44, 1/2, 21/44, 1/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0005/)

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate





## 6. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=7, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=6, threshold=2/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation(6, 131/260, 2, 1, 1, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (6, 131/260, 2, 1, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0006/)

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate





## 7. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=8, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=7, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation(7, 38/75, -1, 0, 0, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (7, 38/75, -1, 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0007/)

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate





## 8. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=9, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=8, threshold=1/2. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation(8, 173/340, 1/2, 167/340, 1/2, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (8, 173/340, 1/2, 167/340, 1/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0008/)

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate





## 9. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=10, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=9, threshold=2/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation(9, 97/190, 2, 1, 1, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (9, 97/190, 2, 1, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0009/)

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate





## 10. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=11, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=10, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation(10, 43/84, -1, 0, 0, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (10, 43/84, -1, 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0010/)

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate





## 11. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=12, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=11, threshold=1/2. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation(11, 231/460, 1/2, 229/460, 1/2, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (11, 231/460, 1/2, 229/460, 1/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0011/)

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate





## 12. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=13, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=12, threshold=2/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation(12, 63/125, 2, 1, 1, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (12, 63/125, 2, 1, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0012/)

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate





## 13. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=14, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=13, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation(13, 91/180, -1, 0, 0, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (13, 91/180, -1, 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0013/)

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate





## 14. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=15, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=14, threshold=1/2. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation(14, 147/290, 1/2, 143/290, 1/2, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (14, 147/290, 1/2, 143/290, 1/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0014/)

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate





## 15. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=16, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=15, threshold=2/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation(15, 63/124, 2, 1, 1, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (15, 63/124, 2, 1, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0015/)

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate





## 16. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=17, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=16, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation(16, 331/660, -1, 0, 0, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (16, 331/660, -1, 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0016/)

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate





## 17. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=18, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=17, threshold=1/2. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation(17, 88/175, 1/2, 87/175, 1/2, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (17, 88/175, 1/2, 87/175, 1/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0017/)

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate





## 18. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=19, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=18, threshold=2/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation(18, 373/740, 2, 1, 1, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (18, 373/740, 2, 1, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0018/)

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate





## 19. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=20, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=19, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation(19, 197/390, -1, 0, 0, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (19, 197/390, -1, 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0019/)

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate





## 20. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=21, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=20, threshold=1/2. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation(20, 83/164, 1/2, 81/164, 1/2, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (20, 83/164, 1/2, 81/164, 1/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0020/)

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate





## 21. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=22, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=21, threshold=2/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation(21, 431/860, 2, 1, 1, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (21, 431/860, 2, 1, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0021/)

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate





## 22. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=23, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=22, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation(22, 113/225, -1, 0, 0, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (22, 113/225, -1, 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0022/)

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate





## 23. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=24, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=23, threshold=1/2. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation(23, 473/940, 1/2, 467/940, 1/2, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (23, 473/940, 1/2, 467/940, 1/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0023/)

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate





## 24. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=25, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=24, threshold=2/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation(24, 247/490, 2, 1, 1, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (24, 247/490, 2, 1, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0024/)

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate





## 25. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=26, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=25, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation(25, 103/204, -1, 0, 0, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (25, 103/204, -1, 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0025/)

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate





## 26. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=2, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=1, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation(1, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (1, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0026/)

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate





## 27. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=3, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=2, threshold=1/2. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation(2, 1/2, 1/50, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (2, 1/2, 1/50, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0027/)

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate





## 28. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=4, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=3, threshold=2/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation(3, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (3, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0028/)

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate





## 29. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=5, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=4, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation(4, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (4, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0029/)

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate





## 30. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=6, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=5, threshold=1/2. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation(5, 1/2, 1/44, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (5, 1/2, 1/44, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0030/)

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate





## 31. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=7, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=6, threshold=2/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation(6, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (6, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0031/)

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate





## 32. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=8, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=7, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation(7, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (7, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0032/)

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate





## 33. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=9, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=8, threshold=1/2. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation(8, 1/2, 3/340, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (8, 1/2, 3/340, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0033/)

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate





## 34. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=10, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=9, threshold=2/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation(9, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (9, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0034/)

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate





## 35. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=11, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=10, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation(10, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (10, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0035/)

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate





## 36. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=12, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=11, threshold=1/2. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation(11, 1/2, 1/460, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (11, 1/2, 1/460, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0036/)

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate





## 37. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=13, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=12, threshold=2/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation(12, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (12, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0037/)

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate





## 38. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=14, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=13, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation(13, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (13, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0038/)

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate





## 39. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=15, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=14, threshold=1/2. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation(14, 1/2, 1/145, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (14, 1/2, 1/145, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0039/)

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate





## 40. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=16, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=15, threshold=2/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation(15, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (15, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0040/)

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate





## 41. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=17, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=16, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation(16, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (16, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0041/)

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate





## 42. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=18, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=17, threshold=1/2. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation(17, 1/2, 1/350, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (17, 1/2, 1/350, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0042/)

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate





## 43. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=19, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=18, threshold=2/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation(18, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (18, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0043/)

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate





## 44. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=20, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=19, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation(19, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (19, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0044/)

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate





## 45. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=21, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=20, threshold=1/2. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation(20, 1/2, 1/164, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (20, 1/2, 1/164, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0045/)

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate





## 46. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=22, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=21, threshold=2/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation(21, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (21, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0046/)

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate





## 47. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=23, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=22, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation(22, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (22, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0047/)

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate





## 48. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=24, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=23, threshold=1/2. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation(23, 1/2, 3/940, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (23, 1/2, 3/940, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0048/)

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate





## 49. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=25, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=24, threshold=2/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation(24, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (24, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0049/)

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate





## 50. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=26, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=25, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation(25, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (25, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0050/)

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate





## 51. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=1/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-12/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation(1/30, -12/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (1/30, -12/7, Yes, CDF_convergence_required_here).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0051/)

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate





## 52. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=2/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-11/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation(1/15, -11/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (1/15, -11/7, Yes, CDF_convergence_required_here).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0052/)

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate





## 53. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=3/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-10/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation(1/10, -10/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (1/10, -10/7, Yes, CDF_convergence_required_here).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0053/)

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate





## 54. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=4/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-9/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation(2/15, -9/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (2/15, -9/7, Yes, CDF_convergence_required_here).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0054/)

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate





## 55. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=5/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-8/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation(1/6, -8/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (1/6, -8/7, Yes, CDF_convergence_required_here).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0055/)

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate





## 56. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=6/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-7/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation(1/5, -1, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (1/5, -1, Yes, CDF_convergence_required_here).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0056/)

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate





## 57. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=7/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-6/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation(7/30, -6/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (7/30, -6/7, Yes, CDF_convergence_required_here).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0057/)

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate





## 58. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=8/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-5/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation(4/15, -5/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (4/15, -5/7, Yes, CDF_convergence_required_here).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0058/)

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate





## 59. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=9/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-4/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation(3/10, -4/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (3/10, -4/7, Yes, CDF_convergence_required_here).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0059/)

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate





## 60. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=10/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-3/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation(1/3, -3/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (1/3, -3/7, Yes, CDF_convergence_required_here).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0060/)

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate





## 61. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=11/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-2/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation(11/30, -2/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (11/30, -2/7, Yes, CDF_convergence_required_here).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0061/)

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate





## 62. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=12/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-1/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation(2/5, -1/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (2/5, -1/7, Yes, CDF_convergence_required_here).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0062/)

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate





## 63. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=13/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=0/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation(13/30, 0, No, limit_CDF_jump_no_requirement_from_definition)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (13/30, 0, No, limit_CDF_jump_no_requirement_from_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0063/)

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate





## 64. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=14/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=1/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation(7/15, 1/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (7/15, 1/7, Yes, CDF_convergence_required_here).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0064/)

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate





## 65. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=15/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=2/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation(1/2, 2/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (1/2, 2/7, Yes, CDF_convergence_required_here).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0065/)

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate





## 66. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=16/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=3/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation(8/15, 3/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (8/15, 3/7, Yes, CDF_convergence_required_here).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0066/)

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate





## 67. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=17/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=4/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation(17/30, 4/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (17/30, 4/7, Yes, CDF_convergence_required_here).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0067/)

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate





## 68. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=18/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=5/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation(3/5, 5/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (3/5, 5/7, Yes, CDF_convergence_required_here).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0068/)

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate





## 69. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=19/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=6/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation(19/30, 6/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (19/30, 6/7, Yes, CDF_convergence_required_here).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0069/)

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate





## 70. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=20/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=7/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation(2/3, 1, No, limit_CDF_jump_no_requirement_from_definition)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (2/3, 1, No, limit_CDF_jump_no_requirement_from_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0070/)

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate





## 71. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=21/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=8/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation(7/10, 8/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (7/10, 8/7, Yes, CDF_convergence_required_here).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0071/)

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate





## 72. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=22/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=9/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation(11/15, 9/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (11/15, 9/7, Yes, CDF_convergence_required_here).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0072/)

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate





## 73. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=23/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=10/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation(23/30, 10/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (23/30, 10/7, Yes, CDF_convergence_required_here).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0073/)

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate





## 74. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=24/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=11/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation(4/5, 11/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (4/5, 11/7, Yes, CDF_convergence_required_here).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0074/)

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate





## 75. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=25/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=12/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation(5/6, 12/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (5/6, 12/7, Yes, CDF_convergence_required_here).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0075/)

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate





## 76. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=2, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=1. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1/2, 1/20, 2, 1, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/20, 2, 1, No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0076/)

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate





## 77. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=3, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=2. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1/2, 1/10, 3, 2, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/10, 3, 2, No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0077/)

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate





## 78. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=4, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=3. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1/2, 3/20, 4, 3, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 3/20, 4, 3, No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0078/)

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate





## 79. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=5, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=4. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

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

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/5, 5, 4, No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0079/)

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate





## 80. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=6, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=5. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1/2, 1/4, 6, 5, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/4, 6, 5, No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0080/)

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate





## 81. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=7, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=6. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1/2, 1/20, 7, 6, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/20, 7, 6, No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0081/)

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate





## 82. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=8, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=7. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1/2, 1/10, 8, 7, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/10, 8, 7, No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0082/)

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate





## 83. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=9, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=8. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1/2, 3/20, 9, 8, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 3/20, 9, 8, No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0083/)

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate





## 84. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=10, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=9. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1/2, 1/5, 10, 9, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/5, 10, 9, No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0084/)

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate





## 85. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=11, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=10. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1/2, 1/4, 11, 10, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/4, 11, 10, No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0085/)

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate





## 86. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=12, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=11. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1/2, 1/20, 12, 11, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/20, 12, 11, No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0086/)

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate





## 87. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=13, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=12. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1/2, 1/10, 13, 12, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/10, 13, 12, No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0087/)

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate





## 88. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=14, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=13. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1/2, 3/20, 14, 13, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 3/20, 14, 13, No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0088/)

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate





## 89. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=15, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=14. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1/2, 1/5, 15, 14, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/5, 15, 14, No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0089/)

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate





## 90. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=16, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=15. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1/2, 1/4, 16, 15, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/4, 16, 15, No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0090/)

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate





## 91. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=17, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=16. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1/2, 1/20, 17, 16, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/20, 17, 16, No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0091/)

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate





## 92. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=18, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=17. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1/2, 1/10, 18, 17, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/10, 18, 17, No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0092/)

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate





## 93. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=19, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=18. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1/2, 3/20, 19, 18, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 3/20, 19, 18, No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0093/)

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate





## 94. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=20, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=19. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1/2, 1/5, 20, 19, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/5, 20, 19, No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0094/)

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate





## 95. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=21, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=20. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1/2, 1/4, 21, 20, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/4, 21, 20, No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0095/)

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate





## 96. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=22, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=21. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1/2, 1/20, 22, 21, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/20, 22, 21, No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0096/)

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate





## 97. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=23, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=22. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1/2, 1/10, 23, 22, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/10, 23, 22, No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0097/)

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate





## 98. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=24, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=23. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1/2, 3/20, 24, 23, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 3/20, 24, 23, No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0098/)

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate





## 99. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=25, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=24. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1/2, 1/5, 25, 24, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/5, 25, 24, No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0099/)

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate





## 100. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=26, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=25. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation(1/2, 1/4, 26, 25, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/4, 26, 25, No, finite_prefix_not_convergence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0100/)

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate



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