### 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)`

Proof

Definitions occuring in Statement :  converges-to: `lim n→∞.x[n] = y` rleq: `x ≤ y` real: `ℝ` nat: `ℕ` uimplies: `b 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: `b supposing a` so_lambda: `λ2x.t[x]` so_apply: `x[s]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` implies: `P `` Q` rleq: `x ≤ y` rnonneg: `rnonneg(x)` le: `A ≤ B` not: `¬A` false: `False` subtype_rel: `A ⊆r B` real: `ℝ` prop: `ℙ`
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

Latex:
\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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