R0041-21097 · q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0022
Original question and givens
Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=22/23, positive c=22 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).
G001 — Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=22/23, positive c=22 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).
Step-by-step derivation
S001 · implies
Before
source operation=monoseq-increasing-limit-certificate;original snapshot and mathematics retained
After
explicit ordered source output fields=[direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate]
Conditions: none
Retain the historical interface marker while listing ordered output fields without JavaScript.Fields and tags are not extra mathematical evidence;fixed No markers retain their claim scope.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
ORIGINAL direction=increasing,L=22/23,c=22>0,shift=2>=1;WHOLE a_n=L+(-1)c/(n+shift) for ALL n>=0
After
for EVERY n>=0,a_(n+1)-a_n=(1)*(22)/((n+2)*(n+1+2));positive denominator;all a_n strictly below L=22/23
Conditions: none
Subtract adjacent symbolic terms using a common denominator.Positivity fixes the sign and gives upper bound L in the increasing case or lower bound L in the decreasing case.This holds for all n,allows negative L,and is not a three-term sample proof.
Step identifiers and dependencies
Rule: monoseq_whole_symbolic_monotonicity_and_matching_bound · Uses: G001, S001
S003 · implies
Before
WHOLE increasing law with ORIGINAL L=22/23,c=22,shift=2
After
for ANY epsilon>0 choose natural N with N+2>22/epsilon;ALL n>=N have |a_n-L|<epsilon;L is supremum but NOT attained because c/(n+shift)>0
Conditions: none
The Archimedean choice controls every later term.Any smaller upper bound or larger lower bound is broken by a term sufficiently close to L,so L is the sup/inf.In the general monotone theorem,sup−epsilon is not an upper bound,which supplies a term;monotonicity then traps the tail.The decreasing proof is dual.
Step identifiers and dependencies
Rule: monoseq_sup_inf_limit_proof_and_nonattainment · Uses: G001, S002
S004 · calculation
Before
monoseq illustrative source n=0 denominator := sum(0, 2)
After
monoseq illustrative source n=0 denominator = 2
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · calculation
Before
monoseq illustrative source n=0 unsigned tail := div(22, 2)
After
monoseq illustrative source n=0 unsigned tail = 11
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · calculation
Before
monoseq illustrative source n=0 signed value := sub(22/23, 11)
After
monoseq illustrative source n=0 signed value = -231/23
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · calculation
Before
monoseq illustrative source n=1 denominator := sum(1, 2)
After
monoseq illustrative source n=1 denominator = 3
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · calculation
Before
monoseq illustrative source n=1 unsigned tail := div(22, 3)
After
monoseq illustrative source n=1 unsigned tail = 22/3
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
S009 · calculation
Before
monoseq illustrative source n=1 signed value := sub(22/23, 22/3)
After
monoseq illustrative source n=1 signed value = -440/69
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · calculation
Before
monoseq illustrative source n=2 denominator := sum(2, 2)
After
monoseq illustrative source n=2 denominator = 4
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · calculation
Before
monoseq illustrative source n=2 unsigned tail := div(22, 4)
After
monoseq illustrative source n=2 unsigned tail = 11/2
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
S012 · calculation
Before
monoseq illustrative source n=2 signed value := sub(22/23, 11/2)
After
monoseq illustrative source n=2 signed value = -209/46
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S011
Result
(increasing, 22/23, 22, 2, [-231/23, -440/69, -209/46], Yes, nondecreasing_bounded_above_supremum_limit)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.