Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[12, 13], coefficients=[13, -14], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=13. Give (n, sup_gap, certificate).
G001 — Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[12, 13], coefficients=[13, -14], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=13. Give (n, sup_gap, certificate).
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.
Keep the historical interface marker in the original snapshot while listing the original ordered fields here.No JavaScript or external table is needed for the main text,and field labels are not mathematical evidence.
original complete domain Fin(2);aligned limit=[12, 13];signed coefficients=[13, -14];shift=2
After
for EVERY n>=0 and EVERY original point x,f_n(x)-limit[x]=coefficient[x]/(n+shift);same positive denominator on the whole domain
Conditions: none
Cancel the same limit[x] from the declared whole-sequence formula,not merely observed samples.Retain original point order;these coefficients and limits are not probabilities.
original n=13;all exact point gaps=[13/15, 14/15];maximum absolute coefficient=14;attaining original point=1
After
EVERY gap<=maximum and selected point attains it;source direct ratio text=14/15;exact reduced value=14/15
Conditions: none
The nonempty finite domain supplies both an upper bound and an attaining point,hence the supremum rather than merely an upper bound.Keep potentially unreduced source ratio bytes while checking their exact value separately.
S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S006 · premise_and_result_audit — Recompute uniform finite maximum gap with exact rational arithmetic, not rounded decimals.
S005 · premise_and_result_audit — Recompute uniform point1 absolute gap with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Recompute uniform point0 absolute gap with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Recompute uniform selected denominator with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
Result
(13, 14/15, exact_finite_domain_supremum)
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.