Nuprl Lemma : const-fun-converges
∀I:Interval. ∀f:ℕ ⟶ ℝ. (f[n]↓ as n→∞ ⇒ λn.f[n]↓ for x ∈ I))
Proof
Definitions occuring in Statement :
fun-converges: λn.f[n; x]↓ for x ∈ I),
interval: Interval,
converges: x[n]↓ as n→∞,
real: ℝ,
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
member: t ∈ T,
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
so_lambda: λ2x y.t[x; y],
rfun: I ⟶ℝ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
rev_implies: P ⇐ Q,
fun-cauchy: λn.f[n; x] is cauchy for x ∈ I,
cauchy: cauchy(n.x[n]),
sq_exists: ∃x:{A| B[x]},
exists: ∃x:A. B[x],
nat_plus: ℕ+,
nat: ℕ,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
guard: {T},
int_upper: {i...},
sq_stable: SqStable(P),
squash: ↓T,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
rleq: x ≤ y,
rnonneg: rnonneg(x),
rneq: x ≠ y
Lemmas referenced :
interval_wf,
converges_wf,
icompact_wf,
nat_plus_subtype_nat,
rleq_wf,
all_wf,
set_wf,
int_upper_wf,
nat_plus_wf,
rabs_wf,
rless_wf,
int_formula_prop_less_lemma,
intformless_wf,
rless-int,
int-to-real_wf,
rdiv_wf,
rsub_wf,
less_than'_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
i-approx_wf,
sq_stable__icompact,
nat_plus_properties,
nat_properties,
le_wf,
int_upper_properties,
int_upper_subtype_nat,
less_than_wf,
le-add-cancel,
add-zero,
add-associates,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
not-lt-2,
false_wf,
decidable__lt,
i-member_wf,
real_wf,
fun-converges-iff-cauchy,
nat_wf,
converges-iff-cauchy
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesisEquality,
hypothesis,
productElimination,
independent_functionElimination,
setEquality,
isectElimination,
setElimination,
rename,
dependent_pairFormation,
dependent_set_memberEquality,
addEquality,
natural_numberEquality,
unionElimination,
independent_pairFormation,
voidElimination,
independent_isectElimination,
isect_memberEquality,
voidEquality,
intEquality,
because_Cache,
minusEquality,
introduction,
imageMemberEquality,
baseClosed,
imageElimination,
int_eqEquality,
computeAll,
independent_pairEquality,
inrFormation,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality
Latex:
\mforall{}I:Interval. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbR{}. (f[n]\mdownarrow{} as n\mrightarrow{}\minfty{} {}\mRightarrow{} \mlambda{}n.f[n]\mdownarrow{} for x \mmember{} I))
Date html generated:
2016_05_18-AM-09_54_25
Last ObjectModification:
2016_01_17-AM-02_53_29
Theory : reals
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