Use the entire nonempty bounded real open interval (20/21, 22/21), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
G001 — Use the entire nonempty bounded real open interval (20/21, 22/21), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).
Definition and scope
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WHOLE original real open interval(20/21,22/21);positive width=2/21;one displayed midpoint=1
After
for EVERY real x in this WHOLE interval,20/21<x<22/21;right=22/21 is an upper bound,left=20/21 is a lower bound;midpoint only witnesses nonemptiness
Conditions: none
The interval contains every real satisfying its strict inequalities,not just the midpoint.Bounds follow from the defining inequalities for every interval point,not from extrema of a sample.
WHOLE interval left L=20/21,right R=22/21;consider ANY real proposed upper bound b<R or lower bound c>L
After
x=(max(L,b)+R)/2 lies in the interval and x>b,so b is NOT upper;y=(L+min(R,c))/2 lies in the interval and y<c,so c is NOT lower;therefore sup=R and inf=L
Conditions: none
For b<R,max(L,b)<R,so their midpoint is an interval point greater than b.The analogous midpoint using min(R,c) lies below c for any c>L.These arguments cover arbitrary real candidates,not only rational samples or a displayed sequence.
original interval conditions L<x<R with L=20/21,R=22/21
After
NEITHER endpoint satisfies both strict inequalities;supremum attained=No,infimum attained=No;for any interior point a midpoint toward each endpoint gives a larger/smaller interior point
Conditions: none
Neither endpoint belongs to the open interval.From any interior point,midpoints toward the endpoints give larger/smaller interior points,so no maximum or minimum exists.Existence of sup/inf is different from attainment.
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S004 · premise_and_result_audit — Recompute supinf one midpoint nonemptiness witness with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Recompute supinf interval midpoint numerator with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Recompute supinf WHOLE open interval positive width 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
(20/21, 22/21, 22/21, 20/21, No, No, supremum_and_infimum_not_attained)
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.