Nuprl Lemma : Raabe-lemma
∀y:ℕ ⟶ ℝ. ∀c:ℝ.
((r0 < c)
⇒ (∀N:ℕ+. ((∀n:{N...}. (r0 < y[n])) ⇒ (∀n:{N...}. (c ≤ (r(n) * ((y[n]/y[n + 1]) - r1)))) ⇒ lim n→∞.y[n] = r0)))
Proof
Definitions occuring in Statement :
converges-to: lim n→∞.x[n] = y,
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rsub: x - y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
int_upper: {i...},
nat_plus: ℕ+,
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
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,
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat_plus: ℕ+,
so_apply: x[s],
nat: ℕ,
int_upper: {i...},
rless: x < y,
sq_exists: ∃x:A [B[x]],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
prop: ℙ,
rneq: x ≠ y,
guard: {T},
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
top: Top,
ge: i ≥ j ,
squash: ↓T,
sq_stable: SqStable(P),
real: ℝ,
subtype_rel: A ⊆r B,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
subtract: n - m,
so_lambda: λ2x.t[x],
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
lelt: i ≤ j < k,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
int_seg: {i..j-},
sq_type: SQType(T),
less_than': less_than'(a;b),
cand: A c∧ B,
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
true: True,
rge: x ≥ y,
less_than: a < b,
rgt: x > y
Latex:
\mforall{}y:\mBbbN{} {}\mrightarrow{} \mBbbR{}. \mforall{}c:\mBbbR{}.
((r0 < c)
{}\mRightarrow{} (\mforall{}N:\mBbbN{}\msupplus{}
((\mforall{}n:\{N...\}. (r0 < y[n]))
{}\mRightarrow{} (\mforall{}n:\{N...\}. (c \mleq{} (r(n) * ((y[n]/y[n + 1]) - r1))))
{}\mRightarrow{} lim n\mrightarrow{}\minfty{}.y[n] = r0)))
Date html generated:
2020_05_20-AM-11_23_02
Last ObjectModification:
2020_01_06-PM-00_48_54
Theory : reals
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