### Nuprl Lemma : not-diverges-converges

`∀[x:ℕ ⟶ ℝ]. (¬(x[n]↓ as n→∞ ∧ n.x[n]↑))`

Proof

Definitions occuring in Statement :  diverges: `n.x[n]↑` converges: `x[n]↓ as n→∞` real: `ℝ` nat: `ℕ` uall: `∀[x:A]. B[x]` so_apply: `x[s]` not: `¬A` and: `P ∧ Q` function: `x:A ⟶ B[x]`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` not: `¬A` implies: `P `` Q` false: `False` and: `P ∧ Q` all: `∀x:A. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` iff: `P `⇐⇒` Q` cauchy: `cauchy(n.x[n])` diverges: `n.x[n]↑` exists: `∃x:A. B[x]` prop: `ℙ` sq_exists: `∃x:{A| B[x]}` guard: `{T}` nat_plus: `ℕ+` uimplies: `b supposing a` rneq: `x ≠ y` or: `P ∨ Q` rev_implies: `P `` Q` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top`
Lemmas referenced :  rless_irreflexivity rless_transitivity1 rsub_wf rabs_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 int-to-real_wf rless_wf small-reciprocal-real real_wf diverges_wf converges_wf and_wf nat_wf converges-iff-cauchy
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation thin sqequalHypSubstitution productElimination lemma_by_obid dependent_functionElimination sqequalRule lambdaEquality applyEquality hypothesisEquality hypothesis independent_functionElimination because_Cache voidElimination isectElimination functionEquality dependent_set_memberEquality natural_numberEquality setElimination rename independent_isectElimination inrFormation unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidEquality independent_pairFormation computeAll

Latex:
\mforall{}[x:\mBbbN{}  {}\mrightarrow{}  \mBbbR{}].  (\mneg{}(x[n]\mdownarrow{}  as  n\mrightarrow{}\minfty{}  \mwedge{}  n.x[n]\muparrow{}))

Date html generated: 2016_05_18-AM-07_39_29
Last ObjectModification: 2016_01_17-AM-02_04_15

Theory : reals

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