Nuprl Lemma : req-from-converges
∀[x:ℕ ⟶ ℝ]. ∀[y:ℝ]. ∀[cvg:lim n→∞.x[n] = y]. (y = cauchy-limit(n.x[n];λk.(cvg (2 * k))))
Proof
Definitions occuring in Statement :
cauchy-limit: cauchy-limit(n.x[n];c),
converges-to: lim n→∞.x[n] = y,
req: x = y,
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
multiply: n * m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
prop: ℙ,
all: ∀x:A. B[x],
uimplies: b supposing a
Lemmas referenced :
converges-cauchy-witness,
sq_stable__req,
cauchy-limit_wf,
nat_wf,
converges-to_wf,
real_wf,
converges-to-cauchy-limit,
unique-limit
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
functionEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination
Latex:
\mforall{}[x:\mBbbN{} {}\mrightarrow{} \mBbbR{}]. \mforall{}[y:\mBbbR{}]. \mforall{}[cvg:lim n\mrightarrow{}\minfty{}.x[n] = y]. (y = cauchy-limit(n.x[n];\mlambda{}k.(cvg (2 * k))))
Date html generated:
2016_10_26-AM-09_16_14
Last ObjectModification:
2016_08_29-PM-06_33_04
Theory : reals
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