Nuprl Lemma : fun-series-converges-on-compact
∀I:Interval. ∀f:ℕ ⟶ I ⟶ℝ.
((∀m:{m:ℕ+| icompact(i-approx(I;m))} . Σn.f[n;x]↓ for x ∈ i-approx(I;m)) ⇒ Σn.f[n;x]↓ for x ∈ I)
Proof
Definitions occuring in Statement :
fun-series-converges: Σn.f[n; x]↓ for x ∈ I,
icompact: icompact(I),
rfun: I ⟶ℝ,
i-approx: i-approx(I;n),
interval: Interval,
nat_plus: ℕ+,
nat: ℕ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
fun-series-converges: Σn.f[n; x]↓ for x ∈ I,
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s]
Lemmas referenced :
fun-converges-on-compact,
all_wf,
nat_plus_wf,
icompact_wf,
i-approx_wf,
fun-converges_wf,
rsum_wf,
i-member-approx,
i-member_wf,
real_wf,
nat_wf,
rfun_wf,
interval_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
because_Cache,
independent_functionElimination,
hypothesis,
isectElimination,
setEquality,
hypothesisEquality,
lambdaEquality,
setElimination,
rename,
applyEquality,
dependent_set_memberEquality,
functionEquality
Latex:
\mforall{}I:Interval. \mforall{}f:\mBbbN{} {}\mrightarrow{} I {}\mrightarrow{}\mBbbR{}.
((\mforall{}m:\{m:\mBbbN{}\msupplus{}| icompact(i-approx(I;m))\} . \mSigma{}n.f[n;x]\mdownarrow{} for x \mmember{} i-approx(I;m)) {}\mRightarrow{} \mSigma{}n.f[n;x]\mdownarrow{} for x \mmember{} I)
Date html generated:
2016_05_18-AM-09_55_16
Last ObjectModification:
2015_12_27-PM-11_08_54
Theory : reals
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