Nuprl Lemma : fun-series-converges-absolutely-converges
∀I:Interval. ∀f:ℕ ⟶ I ⟶ℝ. (Σn.f[n;x]↓ absolutely for x ∈ I ⇒ Σn.f[n;x]↓ for x ∈ I)
Proof
Definitions occuring in Statement :
fun-series-converges-absolutely: Σn.f[n; x]↓ absolutely for x ∈ I,
fun-series-converges: Σn.f[n; x]↓ for x ∈ I,
rfun: I ⟶ℝ,
interval: Interval,
nat: ℕ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
fun-series-converges-absolutely: Σn.f[n; x]↓ absolutely for x ∈ I,
rfun: I ⟶ℝ,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
prop: ℙ,
implies: P ⇒ Q,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
label: ...$L... t
Lemmas referenced :
fun-comparison-test,
rabs_wf,
rfun_wf,
real_wf,
i-member_wf,
nat_wf,
rleq_weakening_equal,
set_wf,
fun-series-converges_wf,
interval_wf
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
lambdaEquality,
sqequalRule,
isectElimination,
applyEquality,
setEquality,
independent_functionElimination,
because_Cache,
independent_isectElimination,
functionEquality
Latex:
\mforall{}I:Interval. \mforall{}f:\mBbbN{} {}\mrightarrow{} I {}\mrightarrow{}\mBbbR{}. (\mSigma{}n.f[n;x]\mdownarrow{} absolutely for x \mmember{} I {}\mRightarrow{} \mSigma{}n.f[n;x]\mdownarrow{} for x \mmember{} I)
Date html generated:
2016_05_18-AM-09_57_09
Last ObjectModification:
2015_12_27-PM-11_07_27
Theory : reals
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