Nuprl Lemma : fun-converges-to-pointwise
∀I:Interval. ∀f:ℕ ⟶ I ⟶ℝ. ∀g:I ⟶ℝ.
(lim n→∞.f[n;x] = λy.g[y] for x ∈ I ⇒ {∀x:ℝ. ((x ∈ I) ⇒ lim n→∞.f[n;x] = g[x])})
Proof
Definitions occuring in Statement :
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,
guard: {T},
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x y.t[x; y],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
and: P ∧ Q,
fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I,
converges-to: lim n→∞.x[n] = y,
sq_exists: ∃x:{A| B[x]},
nat_plus: ℕ+,
nat: ℕ,
uimplies: b supposing a,
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
int_upper: {i...}
Lemmas referenced :
i-member_wf,
real_wf,
fun-converges-to_wf,
nat_wf,
rfun_wf,
interval_wf,
i-approx-containing,
icompact_wf,
i-approx_wf,
i-approx-is-subinterval,
nat_plus_subtype_nat,
le_wf,
all_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_properties,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
nat_plus_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
lambdaEquality,
applyEquality,
functionExtensionality,
setElimination,
rename,
dependent_set_memberEquality,
setEquality,
functionEquality,
dependent_functionElimination,
independent_functionElimination,
productElimination,
because_Cache,
natural_numberEquality,
independent_isectElimination,
inrFormation,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll
Latex:
\mforall{}I:Interval. \mforall{}f:\mBbbN{} {}\mrightarrow{} I {}\mrightarrow{}\mBbbR{}. \mforall{}g:I {}\mrightarrow{}\mBbbR{}.
(lim n\mrightarrow{}\minfty{}.f[n;x] = \mlambda{}y.g[y] for x \mmember{} I {}\mRightarrow{} \{\mforall{}x:\mBbbR{}. ((x \mmember{} I) {}\mRightarrow{} lim n\mrightarrow{}\minfty{}.f[n;x] = g[x])\})
Date html generated:
2016_10_26-AM-11_13_58
Last ObjectModification:
2016_08_27-PM-01_45_23
Theory : reals
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