Nuprl Lemma : converges-to-cauchy-mlimit
∀[X:Type]
∀d:metric(X). ∀cmplt:mcomplete(X with d). ∀x:ℕ ⟶ X. ∀c:mcauchy(d;n.x n). lim n→∞.x n = cauchy-mlimit(cmplt;x;c)
Proof
Definitions occuring in Statement :
cauchy-mlimit: cauchy-mlimit(cmplt;x;c),
mcomplete: mcomplete(M),
mconverges-to: lim n→∞.x[n] = y,
mcauchy: mcauchy(d;n.x[n]),
mk-metric-space: X with d,
metric: metric(X),
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
mcomplete: mcomplete(M),
mk-metric-space: X with d,
cauchy-mlimit: cauchy-mlimit(cmplt;x;c),
member: t ∈ T,
subtype_rel: A ⊆r B,
metric: metric(X),
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
implies: P ⇒ Q,
mconverges: x[n]↓ as n→∞,
exists: ∃x:A. B[x],
pi1: fst(t)
Lemmas referenced :
nat_wf,
subtype_rel_function,
mcauchy_wf,
istype-nat,
mconverges_wf,
subtype_rel_self,
mcomplete_wf,
mk-metric-space_wf,
metric_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
sqequalRule,
cut,
applyEquality,
functionExtensionality,
hypothesisEquality,
functionEquality,
introduction,
extract_by_obid,
hypothesis,
isectElimination,
thin,
setElimination,
rename,
lambdaEquality_alt,
because_Cache,
independent_isectElimination,
inhabitedIsType,
productElimination,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
universeIsType,
functionIsType,
instantiate,
universeEquality
Latex:
\mforall{}[X:Type]
\mforall{}d:metric(X). \mforall{}cmplt:mcomplete(X with d). \mforall{}x:\mBbbN{} {}\mrightarrow{} X. \mforall{}c:mcauchy(d;n.x n).
lim n\mrightarrow{}\minfty{}.x n = cauchy-mlimit(cmplt;x;c)
Date html generated:
2019_10_30-AM-06_43_11
Last ObjectModification:
2019_10_02-AM-10_55_37
Theory : reals
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