### Nuprl Lemma : alt-Riemann-sums-converge

a:ℝ. ∀b:{b:ℝa ≤ b} . ∀f:[a, b] ⟶ℝ. ∀mc:f[x] continuous for x ∈ [a, b].  Riemann-sum-alt(f;a;b;k 1)↓ as k→∞

Proof

Definitions occuring in Statement :  Riemann-sum-alt: Riemann-sum-alt(f;a;b;k) continuous: f[x] continuous for x ∈ I rfun: I ⟶ℝ rccint: [l, u] converges: x[n]↓ as n→∞ rleq: x ≤ y real: so_apply: x[s] all: x:A. B[x] set: {x:A| B[x]}  add: m natural_number: \$n
Definitions unfolded in proof :  all: x:A. B[x] member: t ∈ T so_lambda: λ2x.t[x] uall: [x:A]. B[x] nat_plus: + nat: le: A ≤ B and: P ∧ Q decidable: Dec(P) or: P ∨ Q iff: ⇐⇒ Q not: ¬A rev_implies:  Q implies:  Q false: False prop: uiff: uiff(P;Q) uimplies: supposing a subtract: m subtype_rel: A ⊆B top: Top less_than': less_than'(a;b) true: True so_apply: x[s] label: ...\$L... t rfun: I ⟶ℝ
Rules used in proof :  cut lemma_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation hypothesis sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality sqequalRule lambdaEquality isectElimination dependent_set_memberEquality addEquality setElimination rename natural_numberEquality productElimination unionElimination independent_pairFormation voidElimination independent_functionElimination independent_isectElimination applyEquality isect_memberEquality voidEquality intEquality because_Cache minusEquality setEquality

Latex:
\mforall{}a:\mBbbR{}.  \mforall{}b:\{b:\mBbbR{}|  a  \mleq{}  b\}  .  \mforall{}f:[a,  b]  {}\mrightarrow{}\mBbbR{}.  \mforall{}mc:f[x]  continuous  for  x  \mmember{}  [a,  b].
Riemann-sum-alt(f;a;b;k  +  1)\mdownarrow{}  as  k\mrightarrow{}\minfty{}

Date html generated: 2016_05_18-AM-10_46_31
Last ObjectModification: 2015_12_27-PM-10_47_12

Theory : reals

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