R0041-21026 · q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0051
Step-by-step derivation
S001 · implies
Before
source operation=monoseq-epsilon-witness;original snapshot and mathematics retained
After
explicit ordered source output fields=[direction, limit, epsilon, N, value_at_N, strict_tail_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=1/2,c=1>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)*(1)/((n+2)*(n+1+2));positive denominator;all a_n strictly below L=1/2
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=1/2,c=1,shift=2
After
for ANY epsilon>0 choose natural N with N+2>1/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 c over epsilon := div(1, 1/10)
After
monoseq c over epsilon = 10
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 index multiple := mul(1, 9)
After
monoseq index multiple = 9
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 actual source index := sum(0, 9)
After
monoseq actual source index = 9
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 witness denominator := sum(9, 2)
After
monoseq witness denominator = 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, S006
S008 · calculation
Before
monoseq witness distance := div(1, 11)
After
monoseq witness distance = 1/11
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, S007
S009 · calculation
Before
monoseq predecessor distance := div(1, 10)
After
monoseq predecessor distance = 1/10
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 · implies
Before
monoseq:c=1,shift=2,epsilon=1/10;index(q)=1q+0;c/epsilon=10;floor=10
After
least nonnegative witness=9;actual source index=9;distance=1/11<epsilon;predecessor distance=1/10;no predecessor evaluated when witness0
Conditions: none
Strict distance<epsilon means the denominator exceeds c/epsilon,so its integer threshold is floor(c/epsilon)+1.Round the affine index upward and clamp only the witness to0.Check the predecessor only when positive;equality still fails.
Step identifiers and dependencies
Rule: tail_strict_least_index_not_equality_or_negative_predecessor · Uses: G001, S009
S011 · implies
Before
monoseq:positive c=1,epsilon=1/10,shift=2,stride=1,offset=0;least witness=9
After
for EVERY integer q>=9,denominator=1q+0+2>=witness denominator>0,so 0<distance<=(1/11)<epsilon
Conditions: none
This is an algebraic inequality for every later index,not convergence guessed from one evaluated term.Positive increasing denominators and a fixed positive numerator control the whole tail;the predecessor check proves minimality only.
Step identifiers and dependencies
Rule: tail_all_later_quantified_from_positive_denominator_growth · Uses: G001, S010
S012 · calculation
Before
monoseq at witness N denominator := sum(9, 2)
After
monoseq at witness N denominator = 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, S011
S013 · calculation
Before
monoseq at witness N unsigned tail := div(1, 11)
After
monoseq at witness N unsigned tail = 1/11
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, S012
S014 · calculation
Before
monoseq at witness N signed value := sub(1/2, 1/11)
After
monoseq at witness N signed value = 9/22
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, S013