Nuprl Lemma : fun-converges-on-compact
∀I:Interval. ∀f:ℕ ⟶ I ⟶ℝ.
((∀m:{m:ℕ+| icompact(i-approx(I;m))} . λn.f[n;x]↓ for x ∈ i-approx(I;m))) ⇒ λn.f[n;x]↓ for x ∈ I))
Proof
Definitions occuring in Statement :
fun-converges: λn.f[n; x]↓ for x ∈ I),
icompact: icompact(I),
rfun: I ⟶ℝ,
i-approx: i-approx(I;n),
interval: Interval,
nat_plus: ℕ+,
nat: ℕ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
rfun: I ⟶ℝ,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
fun-cauchy: λn.f[n; x] is cauchy for x ∈ I,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
subinterval: I ⊆ J ,
label: ...$L... t,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
top: Top
Lemmas referenced :
i-approx-approx,
less_than_wf,
i-member-approx,
fun-converges_wf,
all_wf,
icompact_wf,
nat_plus_wf,
subtype_rel_sets,
i-approx_wf,
i-approx-is-subinterval,
nat_wf,
i-member_wf,
real_wf,
rfun_wf,
fun-converges-iff-cauchy
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
isectElimination,
setEquality,
productElimination,
independent_functionElimination,
setElimination,
rename,
because_Cache,
independent_isectElimination,
dependent_set_memberEquality,
functionEquality,
natural_numberEquality,
independent_pairFormation,
introduction,
imageMemberEquality,
baseClosed,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}I:Interval. \mforall{}f:\mBbbN{} {}\mrightarrow{} I {}\mrightarrow{}\mBbbR{}.
((\mforall{}m:\{m:\mBbbN{}\msupplus{}| icompact(i-approx(I;m))\} . \mlambda{}n.f[n;x]\mdownarrow{} for x \mmember{} i-approx(I;m))) {}\mRightarrow{} \mlambda{}n.f[n;x]\mdownarrow{} for x \mmember{} I)\000C)
Date html generated:
2016_05_18-AM-09_54_15
Last ObjectModification:
2016_01_17-AM-02_53_59
Theory : reals
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