### Nuprl Lemma : fun-converges-to-continuous

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

Proof

Definitions occuring in Statement :  fun-converges-to: `lim n→∞.f[n; x] = λy.g[y] for x ∈ I` continuous: `f[x] continuous for x ∈ I` rfun: `I ⟶ℝ` interval: `Interval` nat: `ℕ` uall: `∀[x:A]. B[x]` 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 :  uall: `∀[x:A]. B[x]` implies: `P `` Q` continuous: `f[x] continuous for x ∈ I` all: `∀x:A. B[x]` fun-converges-to: `lim n→∞.f[n; x] = λy.g[y] for x ∈ I` member: `t ∈ T` nat_plus: `ℕ+` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` true: `True` and: `P ∧ Q` prop: `ℙ` exists: `∃x:A. B[x]` subtype_rel: `A ⊆r B` sq_exists: `∃x:{A| B[x]}` cand: `A c∧ B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rfun: `I ⟶ℝ` uimplies: `b supposing a` rneq: `x ≠ y` guard: `{T}` or: `P ∨ Q` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` rless: `x < y` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` label: `...\$L... t` so_apply: `x[s1;s2]` so_lambda: `λ2x y.t[x; y]` rev_uimplies: `rev_uimplies(P;Q)` rge: `x ≥ y` int_upper: `{i...}` real: `ℝ` sq_stable: `SqStable(P)` uiff: `uiff(P;Q)`
Lemmas referenced :  radd-int rdiv_functionality radd-rdiv req_transitivity uiff_transitivity int_term_value_add_lemma itermAdd_wf rleq-int-fractions rabs-difference-symmetry req_weakening radd_functionality rleq_functionality le_wf int_formula_prop_le_lemma intformle_wf decidable__le sq_stable__icompact sq_stable__less_than int_term_value_mul_lemma itermMultiply_wf radd_functionality_wrt_rleq r-triangle-inequality2 rleq_weakening_equal rleq_functionality_wrt_implies radd_wf uimplies_transitivity interval_wf fun-converges-to_wf rfun_wf continuous_wf icompact_wf set_wf nat_plus_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 rless-int rdiv_wf real_wf all_wf int-to-real_wf rless_wf i-approx_wf i-member_wf rsub_wf rabs_wf rleq_wf i-member-approx nat_plus_subtype_nat less_than_wf mul_nat_plus nat_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality cut hypothesis because_Cache lemma_by_obid isectElimination dependent_set_memberEquality natural_numberEquality sqequalRule independent_pairFormation introduction imageMemberEquality baseClosed productElimination applyEquality setElimination rename independent_functionElimination productEquality lambdaEquality functionEquality independent_isectElimination inrFormation unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll setEquality equalityTransitivity equalitySymmetry multiplyEquality addEquality imageElimination

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{}n:\mBbbN{}.  f[n;x]  continuous  for  x  \mmember{}  I)
{}\mRightarrow{}  g[y]  continuous  for  y  \mmember{}  I)

Date html generated: 2016_05_18-AM-09_52_47
Last ObjectModification: 2016_01_17-AM-02_54_59

Theory : reals

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