Nuprl Lemma : fun-ratio-test
∀I:Interval. ∀f:ℕ ⟶ I ⟶ℝ.
((∀n:ℕ. f[n;x] continuous for x ∈ I)
⇒ (∀m:{m:ℕ+| icompact(i-approx(I;m))}
∃c:ℝ. ((r0 ≤ c) ∧ (c < r1) ∧ (∃N:ℕ. ∀n:{N...}. ∀x:{x:ℝ| x ∈ i-approx(I;m)} . (|f[n + 1;x]| ≤ (c * |f[n;x]|)))))
⇒ Σn.f[n;x]↓ absolutely for x ∈ I)
Proof
Definitions occuring in Statement :
fun-series-converges-absolutely: Σn.f[n; x]↓ absolutely for x ∈ I,
continuous: f[x] continuous for x ∈ I,
icompact: icompact(I),
rfun: I ⟶ℝ,
i-approx: i-approx(I;n),
i-member: r ∈ I,
interval: Interval,
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
int_upper: {i...},
nat_plus: ℕ+,
nat: ℕ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃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: $n
Definitions unfolded in proof :
fun-series-converges-absolutely: Σn.f[n; x]↓ absolutely for x ∈ I,
fun-series-converges: Σn.f[n; x]↓ for x ∈ I,
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
rfun: I ⟶ℝ,
uall: ∀[x:A]. B[x],
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
sq_stable: SqStable(P),
nat_plus: ℕ+,
rless: x < y,
sq_exists: ∃x:A [B[x]],
int_upper: {i...},
label: ...$L... t,
cand: A c∧ B,
rleq: x ≤ y,
rnonneg: rnonneg(x),
top: Top,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
guard: {T},
sq_type: SQType(T),
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
real: ℝ,
rge: x ≥ y,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
rabs: |x|,
less_than': less_than'(a;b),
rneq: x ≠ y,
series-converges: Σn.x[n]↓,
series-sum: Σn.x[n] = a,
converges: x[n]↓ as n→∞,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
pointwise-req: x[k] = y[k] for k ∈ [n,m],
subtract: n - m
Latex:
\mforall{}I:Interval. \mforall{}f:\mBbbN{} {}\mrightarrow{} I {}\mrightarrow{}\mBbbR{}.
((\mforall{}n:\mBbbN{}. f[n;x] continuous for x \mmember{} I)
{}\mRightarrow{} (\mforall{}m:\{m:\mBbbN{}\msupplus{}| icompact(i-approx(I;m))\}
\mexists{}c:\mBbbR{}
((r0 \mleq{} c)
\mwedge{} (c < r1)
\mwedge{} (\mexists{}N:\mBbbN{}. \mforall{}n:\{N...\}. \mforall{}x:\{x:\mBbbR{}| x \mmember{} i-approx(I;m)\} . (|f[n + 1;x]| \mleq{} (c * |f[n;x]|)))))
{}\mRightarrow{} \mSigma{}n.f[n;x]\mdownarrow{} absolutely for x \mmember{} I)
Date html generated:
2020_05_20-PM-01_06_42
Last ObjectModification:
2020_01_01-PM-02_27_21
Theory : reals
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