Question 17585: A Long Prefix Still Does Not Replace the All-n Tail

Original question and givens

Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[9, 10], coefficients=[10, -11], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=10, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].

• G001 — Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[9, 10], coefficients=[10, -11], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. prefix_length=10, beginning at n=0. Give (prefix_length,value_rows,sufficient?,finite_prefix_not_pointwise_convergence_certificate), with each row [point,values_in_n_order].

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-pointwise-convergence-of-functions

Step-by-step derivation

same COMPLETE finite domain Fin(2);original aligned limits=[9, 10];signed coefficients=[10, -11];positive shift=2

declared for EVERY n>=0:f_n(x)=limit[x]+coefficient[x]/(n+shift);signed error=f_n(x)-limit[x]=coefficient[x]/(n+shift)

finite prefix point0 index0 denominator := sum(0, 2)

finite prefix point0 index0 denominator = 2

finite prefix point0 index0 signed correction := div(10, 2)

finite prefix point0 index0 signed correction = 5

finite prefix point0 index0 function value := sum(9, 5)

finite prefix point0 index0 function value = 14

finite prefix point0 index0 reverse signed error := sub(14, 9)

finite prefix point0 index0 reverse signed error = 5

finite prefix point0 index0:original point=0;n=0;limit=9;signed coefficient=10

f_n(x)=14;f_n(x)-limit=5;this error field remains SIGNED

finite prefix point0 index1 denominator := sum(1, 2)

finite prefix point0 index1 denominator = 3

finite prefix point0 index1 signed correction := div(10, 3)

finite prefix point0 index1 signed correction = 10/3

finite prefix point0 index1 function value := sum(9, 10/3)

finite prefix point0 index1 function value = 37/3

finite prefix point0 index1 reverse signed error := sub(37/3, 9)

finite prefix point0 index1 reverse signed error = 10/3

finite prefix point0 index1:original point=0;n=1;limit=9;signed coefficient=10

f_n(x)=37/3;f_n(x)-limit=10/3;this error field remains SIGNED

finite prefix point0 index2 denominator := sum(2, 2)

finite prefix point0 index2 denominator = 4

finite prefix point0 index2 signed correction := div(10, 4)

finite prefix point0 index2 signed correction = 5/2

finite prefix point0 index2 function value := sum(9, 5/2)

finite prefix point0 index2 function value = 23/2

finite prefix point0 index2 reverse signed error := sub(23/2, 9)

finite prefix point0 index2 reverse signed error = 5/2

finite prefix point0 index2:original point=0;n=2;limit=9;signed coefficient=10

f_n(x)=23/2;f_n(x)-limit=5/2;this error field remains SIGNED

finite prefix point0 index3 denominator := sum(3, 2)

finite prefix point0 index3 denominator = 5

finite prefix point0 index3 signed correction := div(10, 5)

finite prefix point0 index3 signed correction = 2

finite prefix point0 index3 function value := sum(9, 2)

finite prefix point0 index3 function value = 11

finite prefix point0 index3 reverse signed error := sub(11, 9)

finite prefix point0 index3 reverse signed error = 2

finite prefix point0 index3:original point=0;n=3;limit=9;signed coefficient=10

f_n(x)=11;f_n(x)-limit=2;this error field remains SIGNED

finite prefix point0 index4 denominator := sum(4, 2)

finite prefix point0 index4 denominator = 6

finite prefix point0 index4 signed correction := div(10, 6)

finite prefix point0 index4 signed correction = 5/3

finite prefix point0 index4 function value := sum(9, 5/3)

finite prefix point0 index4 function value = 32/3

finite prefix point0 index4 reverse signed error := sub(32/3, 9)

finite prefix point0 index4 reverse signed error = 5/3

finite prefix point0 index4:original point=0;n=4;limit=9;signed coefficient=10

f_n(x)=32/3;f_n(x)-limit=5/3;this error field remains SIGNED

finite prefix point0 index5 denominator := sum(5, 2)

finite prefix point0 index5 denominator = 7

finite prefix point0 index5 signed correction := div(10, 7)

finite prefix point0 index5 signed correction = 10/7

finite prefix point0 index5 function value := sum(9, 10/7)

finite prefix point0 index5 function value = 73/7

finite prefix point0 index5 reverse signed error := sub(73/7, 9)

finite prefix point0 index5 reverse signed error = 10/7

finite prefix point0 index5:original point=0;n=5;limit=9;signed coefficient=10

f_n(x)=73/7;f_n(x)-limit=10/7;this error field remains SIGNED

finite prefix point0 index6 denominator := sum(6, 2)

finite prefix point0 index6 denominator = 8

finite prefix point0 index6 signed correction := div(10, 8)

finite prefix point0 index6 signed correction = 5/4

finite prefix point0 index6 function value := sum(9, 5/4)

finite prefix point0 index6 function value = 41/4

finite prefix point0 index6 reverse signed error := sub(41/4, 9)

finite prefix point0 index6 reverse signed error = 5/4

finite prefix point0 index6:original point=0;n=6;limit=9;signed coefficient=10

f_n(x)=41/4;f_n(x)-limit=5/4;this error field remains SIGNED

finite prefix point0 index7 denominator := sum(7, 2)

finite prefix point0 index7 denominator = 9

finite prefix point0 index7 signed correction := div(10, 9)

finite prefix point0 index7 signed correction = 10/9

finite prefix point0 index7 function value := sum(9, 10/9)

finite prefix point0 index7 function value = 91/9

finite prefix point0 index7 reverse signed error := sub(91/9, 9)

finite prefix point0 index7 reverse signed error = 10/9

finite prefix point0 index7:original point=0;n=7;limit=9;signed coefficient=10

f_n(x)=91/9;f_n(x)-limit=10/9;this error field remains SIGNED

finite prefix point0 index8 denominator := sum(8, 2)

finite prefix point0 index8 denominator = 10

finite prefix point0 index8 signed correction := div(10, 10)

finite prefix point0 index8 signed correction = 1

finite prefix point0 index8 function value := sum(9, 1)

finite prefix point0 index8 function value = 10

finite prefix point0 index8 reverse signed error := sub(10, 9)

finite prefix point0 index8 reverse signed error = 1

finite prefix point0 index8:original point=0;n=8;limit=9;signed coefficient=10

f_n(x)=10;f_n(x)-limit=1;this error field remains SIGNED

finite prefix point0 index9 denominator := sum(9, 2)

finite prefix point0 index9 denominator = 11

finite prefix point0 index9 signed correction := div(10, 11)

finite prefix point0 index9 signed correction = 10/11

finite prefix point0 index9 function value := sum(9, 10/11)

finite prefix point0 index9 function value = 109/11

finite prefix point0 index9 reverse signed error := sub(109/11, 9)

finite prefix point0 index9 reverse signed error = 10/11

finite prefix point0 index9:original point=0;n=9;limit=9;signed coefficient=10

f_n(x)=109/11;f_n(x)-limit=10/11;this error field remains SIGNED

finite prefix point1 index0 denominator := sum(0, 2)

finite prefix point1 index0 denominator = 2

finite prefix point1 index0 signed correction := div(-11, 2)

finite prefix point1 index0 signed correction = -11/2

finite prefix point1 index0 function value := sum(10, -11/2)

finite prefix point1 index0 function value = 9/2

finite prefix point1 index0 reverse signed error := sub(9/2, 10)

finite prefix point1 index0 reverse signed error = -11/2

finite prefix point1 index0:original point=1;n=0;limit=10;signed coefficient=-11

f_n(x)=9/2;f_n(x)-limit=-11/2;this error field remains SIGNED

finite prefix point1 index1 denominator := sum(1, 2)

finite prefix point1 index1 denominator = 3

finite prefix point1 index1 signed correction := div(-11, 3)

finite prefix point1 index1 signed correction = -11/3

finite prefix point1 index1 function value := sum(10, -11/3)

finite prefix point1 index1 function value = 19/3

finite prefix point1 index1 reverse signed error := sub(19/3, 10)

finite prefix point1 index1 reverse signed error = -11/3

finite prefix point1 index1:original point=1;n=1;limit=10;signed coefficient=-11

f_n(x)=19/3;f_n(x)-limit=-11/3;this error field remains SIGNED

finite prefix point1 index2 denominator := sum(2, 2)

finite prefix point1 index2 denominator = 4

finite prefix point1 index2 signed correction := div(-11, 4)

finite prefix point1 index2 signed correction = -11/4

finite prefix point1 index2 function value := sum(10, -11/4)

finite prefix point1 index2 function value = 29/4

finite prefix point1 index2 reverse signed error := sub(29/4, 10)

finite prefix point1 index2 reverse signed error = -11/4

finite prefix point1 index2:original point=1;n=2;limit=10;signed coefficient=-11

f_n(x)=29/4;f_n(x)-limit=-11/4;this error field remains SIGNED

finite prefix point1 index3 denominator := sum(3, 2)

finite prefix point1 index3 denominator = 5

finite prefix point1 index3 signed correction := div(-11, 5)

finite prefix point1 index3 signed correction = -11/5

finite prefix point1 index3 function value := sum(10, -11/5)

finite prefix point1 index3 function value = 39/5

finite prefix point1 index3 reverse signed error := sub(39/5, 10)

finite prefix point1 index3 reverse signed error = -11/5

finite prefix point1 index3:original point=1;n=3;limit=10;signed coefficient=-11

f_n(x)=39/5;f_n(x)-limit=-11/5;this error field remains SIGNED

finite prefix point1 index4 denominator := sum(4, 2)

finite prefix point1 index4 denominator = 6

finite prefix point1 index4 signed correction := div(-11, 6)

finite prefix point1 index4 signed correction = -11/6

finite prefix point1 index4 function value := sum(10, -11/6)

finite prefix point1 index4 function value = 49/6

finite prefix point1 index4 reverse signed error := sub(49/6, 10)

finite prefix point1 index4 reverse signed error = -11/6

finite prefix point1 index4:original point=1;n=4;limit=10;signed coefficient=-11

f_n(x)=49/6;f_n(x)-limit=-11/6;this error field remains SIGNED

finite prefix point1 index5 denominator := sum(5, 2)

finite prefix point1 index5 denominator = 7

finite prefix point1 index5 signed correction := div(-11, 7)

finite prefix point1 index5 signed correction = -11/7

finite prefix point1 index5 function value := sum(10, -11/7)

finite prefix point1 index5 function value = 59/7

finite prefix point1 index5 reverse signed error := sub(59/7, 10)

finite prefix point1 index5 reverse signed error = -11/7

finite prefix point1 index5:original point=1;n=5;limit=10;signed coefficient=-11

f_n(x)=59/7;f_n(x)-limit=-11/7;this error field remains SIGNED

finite prefix point1 index6 denominator := sum(6, 2)

finite prefix point1 index6 denominator = 8

finite prefix point1 index6 signed correction := div(-11, 8)

finite prefix point1 index6 signed correction = -11/8

finite prefix point1 index6 function value := sum(10, -11/8)

finite prefix point1 index6 function value = 69/8

finite prefix point1 index6 reverse signed error := sub(69/8, 10)

finite prefix point1 index6 reverse signed error = -11/8

finite prefix point1 index6:original point=1;n=6;limit=10;signed coefficient=-11

f_n(x)=69/8;f_n(x)-limit=-11/8;this error field remains SIGNED

finite prefix point1 index7 denominator := sum(7, 2)

finite prefix point1 index7 denominator = 9

finite prefix point1 index7 signed correction := div(-11, 9)

finite prefix point1 index7 signed correction = -11/9

finite prefix point1 index7 function value := sum(10, -11/9)

finite prefix point1 index7 function value = 79/9

finite prefix point1 index7 reverse signed error := sub(79/9, 10)

finite prefix point1 index7 reverse signed error = -11/9

finite prefix point1 index7:original point=1;n=7;limit=10;signed coefficient=-11

f_n(x)=79/9;f_n(x)-limit=-11/9;this error field remains SIGNED

finite prefix point1 index8 denominator := sum(8, 2)

finite prefix point1 index8 denominator = 10

finite prefix point1 index8 signed correction := div(-11, 10)

finite prefix point1 index8 signed correction = -11/10

finite prefix point1 index8 function value := sum(10, -11/10)

finite prefix point1 index8 function value = 89/10

finite prefix point1 index8 reverse signed error := sub(89/10, 10)

finite prefix point1 index8 reverse signed error = -11/10

finite prefix point1 index8:original point=1;n=8;limit=10;signed coefficient=-11

f_n(x)=89/10;f_n(x)-limit=-11/10;this error field remains SIGNED

finite prefix point1 index9 denominator := sum(9, 2)

finite prefix point1 index9 denominator = 11

finite prefix point1 index9 signed correction := div(-11, 11)

finite prefix point1 index9 signed correction = -1

finite prefix point1 index9 function value := sum(10, -1)

finite prefix point1 index9 function value = 9

finite prefix point1 index9 reverse signed error := sub(9, 10)

finite prefix point1 index9 reverse signed error = -1

finite prefix point1 index9:original point=1;n=9;limit=10;signed coefficient=-11

f_n(x)=9;f_n(x)-limit=-1;this error field remains SIGNED

finite observed length=10;complete domain-order rows=[[0, [14, 37/3, 23/2, 11, 32/3, 73/7, 41/4, 91/9, 10, 109/11]], [1, [9/2, 19/3, 29/4, 39/5, 49/6, 59/7, 69/8, 79/9, 89/10, 9]]]

prefix-alone sufficiency=No;finite_prefix_not_pointwise_convergence_certificate

Reverse checks

• S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S101 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S100 · premise_and_result_audit — Recompute finite prefix point1 index9 reverse signed error with exact rational arithmetic, not rounded decimals.
• S099 · premise_and_result_audit — Recompute finite prefix point1 index9 function value with exact rational arithmetic, not rounded decimals.
• S098 · premise_and_result_audit — Recompute finite prefix point1 index9 signed correction with exact rational arithmetic, not rounded decimals.
• S097 · premise_and_result_audit — Recompute finite prefix point1 index9 denominator with exact rational arithmetic, not rounded decimals.
• S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S095 · premise_and_result_audit — Recompute finite prefix point1 index8 reverse signed error with exact rational arithmetic, not rounded decimals.
• S094 · premise_and_result_audit — Recompute finite prefix point1 index8 function value with exact rational arithmetic, not rounded decimals.
• S093 · premise_and_result_audit — Recompute finite prefix point1 index8 signed correction with exact rational arithmetic, not rounded decimals.
• S092 · premise_and_result_audit — Recompute finite prefix point1 index8 denominator with exact rational arithmetic, not rounded decimals.
• S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S090 · premise_and_result_audit — Recompute finite prefix point1 index7 reverse signed error with exact rational arithmetic, not rounded decimals.
• S089 · premise_and_result_audit — Recompute finite prefix point1 index7 function value with exact rational arithmetic, not rounded decimals.
• S088 · premise_and_result_audit — Recompute finite prefix point1 index7 signed correction with exact rational arithmetic, not rounded decimals.
• S087 · premise_and_result_audit — Recompute finite prefix point1 index7 denominator with exact rational arithmetic, not rounded decimals.
• S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S085 · premise_and_result_audit — Recompute finite prefix point1 index6 reverse signed error with exact rational arithmetic, not rounded decimals.
• S084 · premise_and_result_audit — Recompute finite prefix point1 index6 function value with exact rational arithmetic, not rounded decimals.
• S083 · premise_and_result_audit — Recompute finite prefix point1 index6 signed correction with exact rational arithmetic, not rounded decimals.
• S082 · premise_and_result_audit — Recompute finite prefix point1 index6 denominator with exact rational arithmetic, not rounded decimals.
• S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S080 · premise_and_result_audit — Recompute finite prefix point1 index5 reverse signed error with exact rational arithmetic, not rounded decimals.
• S079 · premise_and_result_audit — Recompute finite prefix point1 index5 function value with exact rational arithmetic, not rounded decimals.
• S078 · premise_and_result_audit — Recompute finite prefix point1 index5 signed correction with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Recompute finite prefix point1 index5 denominator with exact rational arithmetic, not rounded decimals.
• S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S075 · premise_and_result_audit — Recompute finite prefix point1 index4 reverse signed error with exact rational arithmetic, not rounded decimals.
• S074 · premise_and_result_audit — Recompute finite prefix point1 index4 function value with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Recompute finite prefix point1 index4 signed correction with exact rational arithmetic, not rounded decimals.
• S072 · premise_and_result_audit — Recompute finite prefix point1 index4 denominator with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S070 · premise_and_result_audit — Recompute finite prefix point1 index3 reverse signed error with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Recompute finite prefix point1 index3 function value with exact rational arithmetic, not rounded decimals.
• S068 · premise_and_result_audit — Recompute finite prefix point1 index3 signed correction with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Recompute finite prefix point1 index3 denominator with exact rational arithmetic, not rounded decimals.
• S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S065 · premise_and_result_audit — Recompute finite prefix point1 index2 reverse signed error with exact rational arithmetic, not rounded decimals.
• S064 · premise_and_result_audit — Recompute finite prefix point1 index2 function value with exact rational arithmetic, not rounded decimals.
• S063 · premise_and_result_audit — Recompute finite prefix point1 index2 signed correction with exact rational arithmetic, not rounded decimals.
• S062 · premise_and_result_audit — Recompute finite prefix point1 index2 denominator with exact rational arithmetic, not rounded decimals.
• S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S060 · premise_and_result_audit — Recompute finite prefix point1 index1 reverse signed error with exact rational arithmetic, not rounded decimals.
• S059 · premise_and_result_audit — Recompute finite prefix point1 index1 function value with exact rational arithmetic, not rounded decimals.
• S058 · premise_and_result_audit — Recompute finite prefix point1 index1 signed correction with exact rational arithmetic, not rounded decimals.
• S057 · premise_and_result_audit — Recompute finite prefix point1 index1 denominator with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S055 · premise_and_result_audit — Recompute finite prefix point1 index0 reverse signed error with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Recompute finite prefix point1 index0 function value with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute finite prefix point1 index0 signed correction with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute finite prefix point1 index0 denominator with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S050 · premise_and_result_audit — Recompute finite prefix point0 index9 reverse signed error with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute finite prefix point0 index9 function value with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute finite prefix point0 index9 signed correction with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute finite prefix point0 index9 denominator with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S045 · premise_and_result_audit — Recompute finite prefix point0 index8 reverse signed error with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute finite prefix point0 index8 function value with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Recompute finite prefix point0 index8 signed correction with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute finite prefix point0 index8 denominator with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S040 · premise_and_result_audit — Recompute finite prefix point0 index7 reverse signed error with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute finite prefix point0 index7 function value with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute finite prefix point0 index7 signed correction with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute finite prefix point0 index7 denominator with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S035 · premise_and_result_audit — Recompute finite prefix point0 index6 reverse signed error with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute finite prefix point0 index6 function value with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute finite prefix point0 index6 signed correction with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute finite prefix point0 index6 denominator with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S030 · premise_and_result_audit — Recompute finite prefix point0 index5 reverse signed error with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute finite prefix point0 index5 function value with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute finite prefix point0 index5 signed correction with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute finite prefix point0 index5 denominator with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S025 · premise_and_result_audit — Recompute finite prefix point0 index4 reverse signed error with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute finite prefix point0 index4 function value with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute finite prefix point0 index4 signed correction with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute finite prefix point0 index4 denominator with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Recompute finite prefix point0 index3 reverse signed error with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute finite prefix point0 index3 function value with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute finite prefix point0 index3 signed correction with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute finite prefix point0 index3 denominator with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Recompute finite prefix point0 index2 reverse signed error with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute finite prefix point0 index2 function value with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute finite prefix point0 index2 signed correction with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute finite prefix point0 index2 denominator with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Recompute finite prefix point0 index1 reverse signed error with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute finite prefix point0 index1 function value with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute finite prefix point0 index1 signed correction with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute finite prefix point0 index1 denominator with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Recompute finite prefix point0 index0 reverse signed error with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute finite prefix point0 index0 function value with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute finite prefix point0 index0 signed correction with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute finite prefix point0 index0 denominator with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(10, [[0, [14, 37/3, 23/2, 11, 32/3, 73/7, 41/4, 91/9, 10, 109/11]], [1, [9/2, 19/3, 29/4, 39/5, 49/6, 59/7, 69/8, 79/9, 89/10, 9]]], No, finite_prefix_not_pointwise_convergence_certificate)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.