Nuprl Lemma : rdiv-factorial-limit-zero-from-bound2
∀x:ℝ. ∀n:ℕ. ((|x| ≤ r(n)) ⇒ lim n→∞.(|x|^n/r((n + 1)!)) = r0)
Proof
Definitions occuring in Statement :
converges-to: lim n→∞.x[n] = y,
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rnexp: x^k1,
int-to-real: r(n),
real: ℝ,
fact: (n)!,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
converges-to: lim n→∞.x[n] = y,
member: t ∈ T,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
nat: ℕ,
prop: ℙ,
sq_exists: ∃x:A [B[x]],
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rneq: x ≠ y,
guard: {T},
rdiv: (x/y),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
top: Top,
rge: x ≥ y,
sq_type: SQType(T),
le: A ≤ B,
less_than: a < b,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtract: n - m,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b
Latex:
\mforall{}x:\mBbbR{}. \mforall{}n:\mBbbN{}. ((|x| \mleq{} r(n)) {}\mRightarrow{} lim n\mrightarrow{}\minfty{}.(|x|\^{}n/r((n + 1)!)) = r0)
Date html generated:
2020_05_20-AM-11_24_05
Last ObjectModification:
2019_12_28-PM-11_48_26
Theory : reals
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