### Nuprl Lemma : Riemann-sums-converge

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

Proof

Definitions occuring in Statement :  Riemann-sum: `Riemann-sum(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: `ℝ` so_apply: `x[s]` all: `∀x:A. B[x]` set: `{x:A| B[x]} ` add: `n + 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: `ℕ+` nat: `ℕ` le: `A ≤ B` and: `P ∧ Q` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` not: `¬A` rev_implies: `P `` Q` implies: `P `` Q` false: `False` prop: `ℙ` uiff: `uiff(P;Q)` uimplies: `b supposing a` subtract: `n - m` subtype_rel: `A ⊆r 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(f;a;b;k  +  1)\mdownarrow{}  as  k\mrightarrow{}\minfty{}

Date html generated: 2016_05_18-AM-10_43_03
Last ObjectModification: 2015_12_27-PM-10_48_22

Theory : reals

Home Index