### 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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