R0041-21427 · q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0052
Original question and givens
Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=2/3, c=2>0, shift=3≥1, parity=1 and epsilon=1/11>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=2/3, c=2>0, shift=3≥1, parity=1 and epsilon=1/11>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=2/3+(-1)^n*(2)/(n+3) for EVERY n>=0;parity=1
After
k(j)=2j+1 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(2, 1/11)
After
bw c over epsilon = 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, S002
S004 · calculation
Before
bw index multiple := mul(2, 10)
After
bw index multiple = 20
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(1, 20)
After
bw actual source index = 21
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(21, 3)
After
bw witness denominator = 24
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(2, 24)
After
bw witness distance = 1/12
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(2, 22)
After
bw predecessor 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 · implies
Before
bw:c=2,shift=3,epsilon=1/11;index(q)=2q+1;c/epsilon=22;floor=22
After
least nonnegative witness=10;actual source index=21;distance=1/12<epsilon;predecessor distance=1/11;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=2,epsilon=1/11,shift=3,stride=2,offset=1;least witness=10
After
for EVERY integer q>=10,denominator=2q+1+3>=witness denominator>0,so 0<distance<=(1/12)<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(21, 3)
After
bw at selected k(J) denominator = 24
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(2, 24)
After
bw at selected k(J) unsigned tail = 1/12
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 := sub(2/3, 1/12)
After
bw at selected k(J) signed value = 7/12
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