Nuprl Lemma : alternating-series-converges
∀x:ℕ ⟶ ℝ. ((∃M:ℕ. ∀n:ℕ. (M < n ⇒ ((r0 ≤ x[n]) ∧ (x[n + 1] ≤ x[n])))) ⇒ lim n→∞.x[n] = r0 ⇒ Σn.-1^n * x[n]↓)
Proof
Definitions occuring in Statement :
series-converges: Σn.x[n]↓,
converges-to: lim n→∞.x[n] = y,
rleq: x ≤ y,
int-rmul: k1 * a,
int-to-real: r(n),
real: ℝ,
fastexp: i^n,
nat: ℕ,
less_than: a < b,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
add: n + m,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
nat: ℕ,
uall: ∀[x:A]. B[x],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
series-converges: Σn.x[n]↓,
series-sum: Σn.x[n] = a,
converges: x[n]↓ as n→∞,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cauchy: cauchy(n.x[n]),
converges-to: lim n→∞.x[n] = y,
sq_exists: ∃x:A [B[x]],
guard: {T},
nat_plus: ℕ+,
rneq: x ≠ y,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
int_upper: {i...},
rge: x ≥ y,
req_int_terms: t1 ≡ t2
Lemmas referenced :
alternating-series-tail-bound,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
converges-to_wf,
istype-nat,
int-to-real_wf,
istype-less_than,
rleq_wf,
real_wf,
converges-iff-cauchy-ext,
rsum_wf,
int-rmul_wf,
fastexp_wf,
int_seg_properties,
int_seg_wf,
imax_wf,
imax_nat,
nat_plus_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
rabs_wf,
rsub_wf,
rdiv_wf,
rless-int,
rless_wf,
nat_plus_wf,
rleq_functionality,
rabs_functionality,
rsum-difference,
req_weakening,
rabs-difference-symmetry,
imax_ub,
itermSubtract_wf,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rabs-of-nonneg
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
dependent_set_memberEquality_alt,
addEquality,
setElimination,
rename,
hypothesis,
natural_numberEquality,
isectElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
because_Cache,
applyEquality,
productIsType,
functionIsType,
minusEquality,
imageElimination,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
equalityIstype,
closedConclusion,
inrFormation_alt,
inlFormation_alt
Latex:
\mforall{}x:\mBbbN{} {}\mrightarrow{} \mBbbR{}
((\mexists{}M:\mBbbN{}. \mforall{}n:\mBbbN{}. (M < n {}\mRightarrow{} ((r0 \mleq{} x[n]) \mwedge{} (x[n + 1] \mleq{} x[n]))))
{}\mRightarrow{} lim n\mrightarrow{}\minfty{}.x[n] = r0
{}\mRightarrow{} \mSigma{}n.-1\^{}n * x[n]\mdownarrow{})
Date html generated:
2019_10_29-AM-10_29_37
Last ObjectModification:
2019_01_30-PM-05_28_43
Theory : reals
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