Nuprl Lemma : cauchy-mlimit-unique
∀[X:Type]. ∀[d:metric(X)]. ∀[cmplt:mcomplete(X with d)]. ∀[x:ℕ ⟶ X]. ∀[c:mcauchy(d;n.x n)]. ∀[z:X].
cauchy-mlimit(cmplt;x;c) ≡ z supposing lim n→∞.x n = z
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,
meq: x ≡ y,
metric: metric(X),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
meq: x ≡ y,
metric: metric(X),
implies: P ⇒ Q,
prop: ℙ
Lemmas referenced :
converges-to-cauchy-mlimit,
m-unique-limit,
istype-nat,
cauchy-mlimit_wf,
meq_inversion,
req_witness,
int-to-real_wf,
mconverges-to_wf,
istype-universe
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
sqequalRule,
lambdaEquality_alt,
applyEquality,
independent_isectElimination,
setElimination,
rename,
natural_numberEquality,
independent_functionElimination,
universeIsType,
isect_memberEquality_alt,
because_Cache,
isectIsTypeImplies,
inhabitedIsType,
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)]. \mforall{}[z:X].
cauchy-mlimit(cmplt;x;c) \mequiv{} z supposing lim n\mrightarrow{}\minfty{}.x n = z
Date html generated:
2019_10_30-AM-06_43_31
Last ObjectModification:
2019_10_02-AM-10_55_55
Theory : reals
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