R0041-21446 · q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0071
Original question and givens
Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=21/22, c=21>0, shift=22≥1, parity=0 and epsilon=1/30>0. The infinite index rule is k(j)=2j+parity for every j≥0. Give (parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate).
G001 — Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=21/22, c=21>0, shift=22≥1, parity=0 and epsilon=1/30>0. The infinite index rule is k(j)=2j+parity for every j≥0. Give (parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate).
Step-by-step derivation
S001 · implies
Before
source operation=bw-oscillatory-tail;original snapshot and mathematics retained
After
explicit ordered source output fields=[parity, index_rule, limit, epsilon, J, value_at_k_J, 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 a_n=21/22+(-1)^n*(21)/(n+22) for EVERY n>=0;parity=0
After
k(j)=2j+0 for ALL j>=0;k(j+1)-k(j)=2>0;selected sign=1;whole sequence bounded by |L|+c/shift
Conditions: none
The parity rule is an infinite strictly increasing index map.Selected signs are fixed and errors equal c/(2j+parity+shift).The whole sequence is bounded;its monotonicity is not a premise for this subsequence certificate.
Step identifiers and dependencies
Rule: bw_parity_whole_law_and_infinite_strict_index · Uses: G001, S001
S003 · calculation
Before
bw c over epsilon := div(21, 1/30)
After
bw c over epsilon = 630
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, S002
S004 · calculation
Before
bw index multiple := mul(2, 305)
After
bw index multiple = 610
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
bw actual source index := sum(0, 610)
After
bw actual source index = 610
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
bw witness denominator := sum(610, 22)
After
bw witness denominator = 632
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
bw witness distance := div(21, 632)
After
bw witness distance = 21/632
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
bw predecessor distance := div(21, 630)
After
bw predecessor distance = 1/30
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 · implies
Before
bw:c=21,shift=22,epsilon=1/30;index(q)=2q+0;c/epsilon=630;floor=630
After
least nonnegative witness=305;actual source index=610;distance=21/632<epsilon;predecessor distance=1/30;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, S008
S010 · implies
Before
bw:positive c=21,epsilon=1/30,shift=22,stride=2,offset=0;least witness=305
After
for EVERY integer q>=305,denominator=2q+0+22>=witness denominator>0,so 0<distance<=(21/632)<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, S009
S011 · calculation
Before
bw at selected k(J) denominator := sum(610, 22)
After
bw at selected k(J) denominator = 632
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
bw at selected k(J) unsigned tail := div(21, 632)
After
bw at selected k(J) unsigned tail = 21/632
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
bw at selected k(J) signed value := sum(21/22, 21/632)
After
bw at selected k(J) signed value = 6867/6952
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