Nuprl Lemma : converges-cauchy-witness
∀[x:ℕ ⟶ ℝ]. ∀[y:ℝ]. ∀[cvg:lim n→∞.x[n] = y]. (λk.(cvg (2 * k)) ∈ cauchy(n.x[n]))
Proof
Definitions occuring in Statement :
cauchy: cauchy(n.x[n]),
converges-to: lim n→∞.x[n] = y,
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
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,
converges: x[n]↓ as n→∞,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
converges-iff-cauchy-ext,
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
top: Top,
rev_implies: P ⇐ Q,
pi1: fst(t)
Lemmas referenced :
converges-to_wf,
nat_wf,
real_wf,
converges-iff-cauchy-ext,
all_wf,
iff_wf,
converges_wf,
cauchy_wf,
pi1_wf_top,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
dependent_pairEquality,
hypothesisEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
because_Cache,
functionEquality,
instantiate,
lambdaFormation,
productElimination,
independent_pairEquality,
voidElimination,
voidEquality,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[x:\mBbbN{} {}\mrightarrow{} \mBbbR{}]. \mforall{}[y:\mBbbR{}]. \mforall{}[cvg:lim n\mrightarrow{}\minfty{}.x[n] = y]. (\mlambda{}k.(cvg (2 * k)) \mmember{} cauchy(n.x[n]))
Date html generated:
2016_10_26-AM-09_16_06
Last ObjectModification:
2016_08_29-PM-06_26_26
Theory : reals
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