Nuprl Lemma : rdiv-factorial-limit-zero
∀x:ℝ. lim n→∞.(|x|^n/r((n)!)) = r0
Proof
Definitions occuring in Statement :
converges-to: lim n→∞.x[n] = y,
rdiv: (x/y),
rabs: |x|,
rnexp: x^k1,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
natural_number: $n,
fact: (n)!
Definitions unfolded in proof :
all: ∀x:A. B[x],
converges-to: lim n→∞.x[n] = y,
member: t ∈ T,
exists: ∃x:A. B[x],
sq_exists: ∃x:{A| B[x]},
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
nat: ℕ,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
so_apply: x[s],
rdiv: (x/y),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
real: ℝ,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
rge: x ≥ y,
sq_type: SQType(T),
squash: ↓T,
less_than: a < b,
subtract: n - m,
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
nequal: a ≠ b ∈ T
Lemmas referenced :
r-archimedean-rabs,
expfact-property,
nat_plus_wf,
real_wf,
nat_plus_subtype_nat,
le_wf,
nat_wf,
all_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
rnexp_wf,
int-to-real_wf,
fact_wf,
rless-int,
nat_properties,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
rless_wf,
rmul_wf,
rinv_wf2,
rleq_functionality,
rabs_functionality,
req_transitivity,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rinv-mul-as-rdiv,
rinv-as-rdiv,
rabs-of-nonneg,
rmul-rdiv-cancel2,
req_weakening,
rmul-int,
uiff_transitivity,
less_than_wf,
exp_wf_nat_plus,
decidable__le,
rmul_preserves_rleq,
zero-rleq-rabs,
rnexp-nonneg,
zero-mul,
rnexp-rleq,
rmul-int-rdiv,
rmul_comm,
rnexp-int,
exp_wf2,
less_than'_wf,
int_formula_prop_le_lemma,
intformle_wf,
rleq-int,
rmul_preserves_rleq2,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
int_subtype_base,
subtype_base_sq,
decidable__equal_int,
int_formula_prop_eq_lemma,
intformeq_wf,
exp_preserves_lt,
false_wf,
int_term_value_mul_lemma,
multiply-is-int-iff,
exp_wf4,
mul_preserves_le,
fact-bound,
int_term_value_subtract_lemma,
subtract_wf,
int_term_value_add_lemma,
itermAdd_wf,
add-zero,
ge_wf,
exp_step,
add-associates,
add-commutes,
add-swap,
mul-swap,
le_weakening,
le_functionality,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
fact_unroll,
trivial-int-eq1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
hypothesis,
dependent_set_memberFormation,
applyEquality,
sqequalRule,
rename,
isectElimination,
setElimination,
lambdaEquality,
functionEquality,
because_Cache,
independent_isectElimination,
inrFormation,
independent_functionElimination,
natural_numberEquality,
dependent_set_memberEquality,
unionElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
voidElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
multiplyEquality,
axiomEquality,
minusEquality,
independent_pairEquality,
isect_memberFormation,
cumulativity,
instantiate,
imageElimination,
baseClosed,
closedConclusion,
baseApply,
promote_hyp,
pointwiseFunctionality,
addEquality,
intWeakElimination,
equalityElimination
Latex:
\mforall{}x:\mBbbR{}. lim n\mrightarrow{}\minfty{}.(|x|\^{}n/r((n)!)) = r0
Date html generated:
2017_10_03-AM-09_27_16
Last ObjectModification:
2017_07_28-AM-07_47_13
Theory : reals
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