Nuprl Lemma : ratio-test
∀x:ℕ ⟶ ℝ. ∀N:ℕ.
((∀c:{c:ℝ| (r0 ≤ c) ∧ (c < r1)} . ((∀n:{N...}. (|x[n + 1]| ≤ (c * |x[n]|))) ⇒ Σn.x[n]↓))
∧ (∀c:{c:ℝ| r1 < c} . ((∀n:{N...}. ((c * |x[n]|) < |x[n + 1]|)) ⇒ Σn.x[n]↑)))
Proof
Definitions occuring in Statement :
series-diverges: Σn.x[n]↑,
series-converges: Σn.x[n]↓,
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
int_upper: {i...},
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
so_apply: x[s],
int_upper: {i...},
ge: i ≥ j ,
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,
prop: ℙ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
sq_type: SQType(T),
guard: {T},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
sq_stable: SqStable(P),
squash: ↓T,
rge: x ≥ y,
so_lambda: λ2x.t[x],
true: True,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
series-converges: Σn.x[n]↓,
rneq: x ≠ y,
less_than': less_than'(a;b),
rgt: x > y,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
real: ℝ,
subtract: n - m
Latex:
\mforall{}x:\mBbbN{} {}\mrightarrow{} \mBbbR{}. \mforall{}N:\mBbbN{}.
((\mforall{}c:\{c:\mBbbR{}| (r0 \mleq{} c) \mwedge{} (c < r1)\} . ((\mforall{}n:\{N...\}. (|x[n + 1]| \mleq{} (c * |x[n]|))) {}\mRightarrow{} \mSigma{}n.x[n]\mdownarrow{}))
\mwedge{} (\mforall{}c:\{c:\mBbbR{}| r1 < c\} . ((\mforall{}n:\{N...\}. ((c * |x[n]|) < |x[n + 1]|)) {}\mRightarrow{} \mSigma{}n.x[n]\muparrow{})))
Date html generated:
2020_05_20-AM-11_22_18
Last ObjectModification:
2020_01_06-PM-00_26_02
Theory : reals
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