Nuprl Lemma : fun-converges-converges-to
∀I:Interval. ∀f:ℕ ⟶ I ⟶ℝ. ∀g:I ⟶ℝ.
((∀x:{x:ℝ| x ∈ I} . lim n→∞.f[n;x] = g[x]) ⇒ λn.f[n;x]↓ for x ∈ I) ⇒ lim n→∞.f[n;x] = λx.g[x] for x ∈ I)
Proof
Definitions occuring in Statement :
fun-converges: λn.f[n; x]↓ for x ∈ I),
fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
converges-to: lim n→∞.x[n] = y,
real: ℝ,
nat: ℕ,
so_apply: x[s1;s2],
so_apply: x[s],
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,
fun-converges: λn.f[n; x]↓ for x ∈ I),
exists: ∃x:A. B[x],
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,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
label: ...$L... t,
sq_stable: SqStable(P),
squash: ↓T,
uimplies: b supposing a
Lemmas referenced :
fun-converges-to_functionality2,
nat_wf,
i-member_wf,
real_wf,
set_wf,
fun-converges_wf,
rfun_wf,
all_wf,
converges-to_wf,
interval_wf,
fun-converges-to-pointwise,
sq_stable__i-member,
unique-limit
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
rename,
cut,
hypothesis,
addLevel,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
because_Cache,
setElimination,
dependent_set_memberEquality,
isectElimination,
setEquality,
functionEquality,
independent_functionElimination,
levelHypothesis,
imageMemberEquality,
baseClosed,
imageElimination,
independent_isectElimination
Latex:
\mforall{}I:Interval. \mforall{}f:\mBbbN{} {}\mrightarrow{} I {}\mrightarrow{}\mBbbR{}. \mforall{}g:I {}\mrightarrow{}\mBbbR{}.
((\mforall{}x:\{x:\mBbbR{}| x \mmember{} I\} . lim n\mrightarrow{}\minfty{}.f[n;x] = g[x])
{}\mRightarrow{} \mlambda{}n.f[n;x]\mdownarrow{} for x \mmember{} I)
{}\mRightarrow{} lim n\mrightarrow{}\minfty{}.f[n;x] = \mlambda{}x.g[x] for x \mmember{} I)
Date html generated:
2016_10_26-AM-11_14_13
Last ObjectModification:
2016_08_27-PM-08_25_21
Theory : reals
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