Nuprl Lemma : rleq-limit-constant

[x:ℝ]. ∀[y:ℕ ⟶ ℝ]. ∀[a:ℝ].  (a ≤ x) supposing ((∀n:ℕ(y[n] ≤ x)) and lim n→∞.y[n] a)


Definitions occuring in Statement :  converges-to: lim n→∞.x[n] y rleq: x ≤ y real: nat: uimplies: supposing a uall: [x:A]. B[x] so_apply: x[s] all: x:A. B[x] function: x:A ⟶ B[x]
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a so_lambda: λ2x.t[x] so_apply: x[s] all: x:A. B[x] iff: ⇐⇒ Q and: P ∧ Q rev_implies:  Q implies:  Q rleq: x ≤ y rnonneg: rnonneg(x) le: A ≤ B not: ¬A false: False subtype_rel: A ⊆B real: prop:
Lemmas referenced :  rleq-limit nat_wf constant-limit req_weakening less_than'_wf rsub_wf real_wf nat_plus_wf all_wf rleq_wf converges-to_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality sqequalRule lambdaEquality hypothesis independent_isectElimination dependent_functionElimination because_Cache productElimination independent_functionElimination independent_pairEquality applyEquality setElimination rename minusEquality natural_numberEquality axiomEquality equalityTransitivity equalitySymmetry isect_memberEquality functionEquality voidElimination

\mforall{}[x:\mBbbR{}].  \mforall{}[y:\mBbbN{}  {}\mrightarrow{}  \mBbbR{}].  \mforall{}[a:\mBbbR{}].    (a  \mleq{}  x)  supposing  ((\mforall{}n:\mBbbN{}.  (y[n]  \mleq{}  x))  and  lim  n\mrightarrow{}\minfty{}.y[n]  =  a)

Date html generated: 2016_05_18-AM-07_53_21
Last ObjectModification: 2015_12_28-AM-01_07_28

Theory : reals

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