Nuprl Lemma : power-series-converges-everywhere-to
∀a:ℕ ⟶ ℝ. ∀b:ℝ. ∀g:ℝ ⟶ ℝ.
((∀x:ℝ. Σi.a[i] * x - b^i = g[x])
⇒ (∀r:{r:ℝ| r0 < r} . ∃N:ℕ. ∀n:{N...}. (|a[n + 1]| ≤ (|a[n]|/r)))
⇒ lim n→∞.Σ{a[i] * x - b^i | 0≤i≤n} = λx.g[x] for x ∈ (-∞, ∞))
Proof
Definitions occuring in Statement :
fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I,
riiint: (-∞, ∞),
series-sum: Σn.x[n] = a,
rsum: Σ{x[k] | n≤k≤m},
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rnexp: x^k1,
rsub: x - y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
int_upper: {i...},
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃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],
member: t ∈ T,
implies: P ⇒ Q,
fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I,
i-approx: i-approx(I;n),
riiint: (-∞, ∞),
uall: ∀[x:A]. B[x],
prop: ℙ,
exists: ∃x:A. B[x],
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),
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,
true: True,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
subinterval: I ⊆ J ,
cand: A c∧ B,
rge: x ≥ y,
rless: x < y,
sq_exists: ∃x:A [B[x]],
rfun: I ⟶ℝ,
i-member: r ∈ I,
rooint: (l, u),
icompact: icompact(I),
i-nonvoid: i-nonvoid(I),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b
Latex:
\mforall{}a:\mBbbN{} {}\mrightarrow{} \mBbbR{}. \mforall{}b:\mBbbR{}. \mforall{}g:\mBbbR{} {}\mrightarrow{} \mBbbR{}.
((\mforall{}x:\mBbbR{}. \mSigma{}i.a[i] * x - b\^{}i = g[x])
{}\mRightarrow{} (\mforall{}r:\{r:\mBbbR{}| r0 < r\} . \mexists{}N:\mBbbN{}. \mforall{}n:\{N...\}. (|a[n + 1]| \mleq{} (|a[n]|/r)))
{}\mRightarrow{} lim n\mrightarrow{}\minfty{}.\mSigma{}\{a[i] * x - b\^{}i | 0\mleq{}i\mleq{}n\} = \mlambda{}x.g[x] for x \mmember{} (-\minfty{}, \minfty{}))
Date html generated:
2020_05_20-PM-01_07_43
Last ObjectModification:
2020_01_06-PM-00_29_22
Theory : reals
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