Nuprl Lemma : power-series-converges
∀a:ℕ ⟶ ℝ. ∀b:ℝ. ∀r:{r:ℝ| r0 < r} . ∀N:ℕ.
((∀n:{N...}. (|a[n + 1]| ≤ (|a[n]|/r))) ⇒ Σn.a[n] * x - b^n↓ absolutely for x ∈ (b - r, b + r))
Proof
Definitions occuring in Statement :
fun-series-converges-absolutely: Σn.f[n; x]↓ absolutely for x ∈ I,
rooint: (l, u),
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rnexp: x^k1,
rsub: x - y,
rmul: a * b,
radd: 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,
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],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x y.t[x; y],
rfun: I ⟶ℝ,
so_apply: x[s],
prop: ℙ,
so_apply: x[s1;s2],
i-approx: i-approx(I;n),
rooint: (l, u),
nat: ℕ,
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,
and: P ∧ Q,
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
squash: ↓T,
rneq: x ≠ y,
guard: {T},
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
uiff: uiff(P;Q),
rless: x < y,
sq_exists: ∃x:A [B[x]],
req_int_terms: t1 ≡ t2,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
rev_uimplies: rev_uimplies(P;Q),
i-member: r ∈ I,
rccint: [l, u],
rdiv: (x/y),
rge: x ≥ y,
subtract: n - m
Latex:
\mforall{}a:\mBbbN{} {}\mrightarrow{} \mBbbR{}. \mforall{}b:\mBbbR{}. \mforall{}r:\{r:\mBbbR{}| r0 < r\} . \mforall{}N:\mBbbN{}.
((\mforall{}n:\{N...\}. (|a[n + 1]| \mleq{} (|a[n]|/r))) {}\mRightarrow{} \mSigma{}n.a[n] * x - b\^{}n\mdownarrow{} absolutely for x \mmember{} (b - r, b + r))
Date html generated:
2020_05_20-PM-01_07_10
Last ObjectModification:
2019_12_14-PM-02_55_55
Theory : reals
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