### 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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