Nuprl Lemma : fun-series-converges-to-everywhere
∀f:ℕ ⟶ ℝ ⟶ ℝ
((∀n:ℕ. ∀x,y:ℝ. ((x = y) ⇒ (f[n;x] = f[n;y])))
⇒ (∀m:ℕ+. ∃c:ℝ. ((r0 ≤ c) ∧ (c < r1) ∧ (∃N:ℕ. ∀n:{N...}. ∀x:{x:ℝ| |x| ≤ r(m)} . (|f[n + 1;x]| ≤ (c * |f[n;x]|)))))
⇒ (∀g:ℝ ⟶ ℝ. ((∀x:ℝ. lim n→∞.Σ{f[i;x] | 0≤i≤n} = g[x]) ⇒ lim n→∞.Σ{f[i;x] | 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: (-∞, ∞),
rsum: Σ{x[k] | n≤k≤m},
converges-to: lim n→∞.x[n] = y,
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
int_upper: {i...},
nat_plus: ℕ+,
nat: ℕ,
so_apply: x[s1;s2],
so_apply: x[s],
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 :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
rfun: I ⟶ℝ,
so_apply: x[s1;s2],
uall: ∀[x:A]. B[x],
prop: ℙ,
fun-series-converges: Σn.f[n; x]↓ for x ∈ I,
nat: ℕ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
nat_plus: ℕ+,
int_upper: {i...},
guard: {T},
rless: x < y,
sq_exists: ∃x:{A| B[x]},
real: ℝ,
sq_stable: SqStable(P),
squash: ↓T,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top
Lemmas referenced :
fun-ratio-test-everywhere,
fun-series-converges-absolutely-converges,
riiint_wf,
nat_wf,
real_wf,
i-member_wf,
fun-converges-converges-to,
rsum_wf,
int_seg_wf,
set_wf,
all_wf,
converges-to_wf,
int_seg_subtype_nat,
false_wf,
nat_plus_wf,
exists_wf,
rleq_wf,
int-to-real_wf,
rless_wf,
int_upper_wf,
rabs_wf,
int_upper_properties,
nat_properties,
sq_stable__less_than,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
rmul_wf,
int_upper_subtype_nat,
req_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
setElimination,
rename,
setEquality,
isectElimination,
because_Cache,
natural_numberEquality,
addEquality,
independent_isectElimination,
independent_pairFormation,
functionEquality,
productEquality,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll
Latex:
\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbR{} {}\mrightarrow{} \mBbbR{}
((\mforall{}n:\mBbbN{}. \mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} (f[n;x] = f[n;y])))
{}\mRightarrow{} (\mforall{}m:\mBbbN{}\msupplus{}
\mexists{}c:\mBbbR{}
((r0 \mleq{} c)
\mwedge{} (c < r1)
\mwedge{} (\mexists{}N:\mBbbN{}. \mforall{}n:\{N...\}. \mforall{}x:\{x:\mBbbR{}| |x| \mleq{} r(m)\} . (|f[n + 1;x]| \mleq{} (c * |f[n;x]|)))))
{}\mRightarrow{} (\mforall{}g:\mBbbR{} {}\mrightarrow{} \mBbbR{}
((\mforall{}x:\mBbbR{}. lim n\mrightarrow{}\minfty{}.\mSigma{}\{f[i;x] | 0\mleq{}i\mleq{}n\} = g[x])
{}\mRightarrow{} lim n\mrightarrow{}\minfty{}.\mSigma{}\{f[i;x] | 0\mleq{}i\mleq{}n\} = \mlambda{}x.g[x] for x \mmember{} (-\minfty{}, \minfty{}))))
Date html generated:
2016_10_26-AM-11_14_44
Last ObjectModification:
2016_08_28-PM-02_18_31
Theory : reals
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