# Question 17557: Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers [Return to catalogue page 88](https://ocme.evemisslab.com/en/questions/page/88/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations/#q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0057) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0034-17557 · q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0057 ## Original question and givens Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[6, 7], coefficients=[7, -8], 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. Positive epsilon=1/16. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order. - G001 — Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[6, 7], coefficients=[7, -8], 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. Positive epsilon=1/16. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain 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](https://ocme.evemisslab.com/en/mko/mko-pointwise-convergence-of-functions/) ## Step-by-step derivation S001 · impliesBefore same COMPLETE finite domain Fin(2);original aligned limits=[6, 7];signed coefficients=[7, -8];positive shift=2 After 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) Conditions: none The complete common finite domain and all-n formula are declared premises,not extrapolated from samples.Function values and errors are not probabilities;they may be signed or exceed1.Do not clip or shift indices. Step identifiers and dependenciesRule: pointwise_explicit_common_domain_all_n_tail_not_probability · Uses: G001 S002 · calculationBefore finite-domain witness point0 absolute coefficient cross product := mul(7, 16) After finite-domain witness point0 absolute coefficient cross product = 112 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore finite-domain witness point0 strict reciprocal threshold := div(112, 1) After finite-domain witness point0 strict reciprocal threshold = 112 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore finite-domain witness point0 raw strict integer cutoff := sum(112, -2, 1) After finite-domain witness point0 raw strict integer cutoff = 111 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore finite-domain witness point0:point=0;SIGNED coefficient=7;absolute coefficient=7;positive epsilon=1/16;shift=2 After threshold=112;floor=112;least nonnegative strict cutoff N=111 Conditions: none The strict inequality is N+s>|c|b/a,so take its floor plus1 minus shift and clamp only at0.Equality is insufficient and a zero coefficient permits N=0.No probability≤1 guard belongs here. Step identifiers and dependenciesRule: pointwise_least_strict_cutoff_floor_not_probability_guard · Uses: G001, S004 S006 · calculationBefore finite-domain witness point0 cutoff denominator := sum(111, 2) After finite-domain witness point0 cutoff denominator = 113 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore finite-domain witness point0 absolute error at cutoff := div(7, 113) After finite-domain witness point0 absolute error at cutoff = 7/113 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore finite-domain witness point0 previous denominator := sum(110, 2) After finite-domain witness point0 previous denominator = 112 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore finite-domain witness point0 previous absolute error := div(7, 112) After finite-domain witness point0 previous absolute error = 1/16 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore finite-domain witness point0:current N=111;current error=7/113;previous error=1/16;epsilon=1/16 After current strict bound holds;previous fails;all smaller indices fail by monotonicity Conditions: none The preceding error is still≥epsilon.As denominators increase with n,every smaller nonnegative index also fails the strict bound. Step identifiers and dependenciesRule: pointwise_minimality_previous_index_strict_threshold · Uses: G001, S009 S011 · impliesBefore finite-domain witness point0:declared exact tail;positive denominators increase for all n>=N=111 After for EVERY later n,absolute error<=bound=7/113|c|b/a,so take its floor plus1 minus shift and clamp only at0.Equality is insufficient and a zero coefficient permits N=0.No probability≤1 guard belongs here. Step identifiers and dependenciesRule: pointwise_least_strict_cutoff_floor_not_probability_guard · Uses: G001, S014 S016 · calculationBefore finite-domain witness point1 cutoff denominator := sum(127, 2) After finite-domain witness point1 cutoff denominator = 129 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore finite-domain witness point1 absolute error at cutoff := div(8, 129) After finite-domain witness point1 absolute error at cutoff = 8/129 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore finite-domain witness point1 previous denominator := sum(126, 2) After finite-domain witness point1 previous denominator = 128 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore finite-domain witness point1 previous absolute error := div(8, 128) After finite-domain witness point1 previous absolute error = 1/16 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · impliesBefore finite-domain witness point1:current N=127;current error=8/129;previous error=1/16;epsilon=1/16 After current strict bound holds;previous fails;all smaller indices fail by monotonicity Conditions: none The preceding error is still≥epsilon.As denominators increase with n,every smaller nonnegative index also fails the strict bound. Step identifiers and dependenciesRule: pointwise_minimality_previous_index_strict_threshold · Uses: G001, S019 S021 · impliesBefore finite-domain witness point1:declared exact tail;positive denominators increase for all n>=N=127 After for EVERY later n,absolute error<=bound=8/129