### Nuprl Lemma : fun-converges-to_wf

`∀[I:Interval]. ∀[f:ℕ ⟶ I ⟶ℝ]. ∀[g:I ⟶ℝ].  (lim n→∞.f[n;x] = λy.g[y] for x ∈ I ∈ ℙ)`

Proof

Definitions occuring in Statement :  fun-converges-to: `lim n→∞.f[n; x] = λy.g[y] for x ∈ I` rfun: `I ⟶ℝ` interval: `Interval` nat: `ℕ` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` so_apply: `x[s]` member: `t ∈ T` function: `x:A ⟶ B[x]`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` implies: `P `` Q` uall: `∀[x:A]. B[x]` prop: `ℙ` fun-converges-to: `lim n→∞.f[n; x] = λy.g[y] for x ∈ I` so_lambda: `λ2x.t[x]` nat_plus: `ℕ+` so_apply: `x[s1;s2]` subtype_rel: `A ⊆r B` rfun: `I ⟶ℝ` so_apply: `x[s]` uimplies: `b supposing a` rneq: `x ≠ y` guard: `{T}` or: `P ∨ Q` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` int_upper: `{i...}` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top`
Lemmas referenced :  interval_wf nat_wf rfun_wf rless_wf int_formula_prop_wf int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_less_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermConstant_wf intformless_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__lt nat_plus_properties int_upper_properties rless-int int-to-real_wf rdiv_wf nat_plus_subtype_nat int_upper_subtype_nat rsub_wf rabs_wf rleq_wf int_upper_wf real_wf exists_wf i-approx_wf icompact_wf nat_plus_wf all_wf i-member_wf i-member-approx
Rules used in proof :  cut lemma_by_obid sqequalHypSubstitution sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity dependent_functionElimination thin hypothesisEquality independent_functionElimination hypothesis dependent_set_memberEquality because_Cache isectElimination isect_memberFormation introduction sqequalRule setEquality lambdaEquality lambdaFormation setElimination rename applyEquality natural_numberEquality independent_isectElimination inrFormation productElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll axiomEquality equalityTransitivity equalitySymmetry functionEquality

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  \mmember{}  \mBbbP{})

Date html generated: 2016_05_18-AM-09_52_33
Last ObjectModification: 2016_01_17-AM-02_53_18

Theory : reals

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