Nuprl Lemma : constant-rleq-limit
∀[x:ℝ]. ∀[y:ℕ ⟶ ℝ]. ∀[a:ℝ]. (x ≤ a) supposing ((∀n:ℕ. (x ≤ y[n])) 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,
sqequalRule,
lambdaEquality,
hypothesisEquality,
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{}]. (x \mleq{} a) supposing ((\mforall{}n:\mBbbN{}. (x \mleq{} y[n])) and lim n\mrightarrow{}\minfty{}.y[n] = a)
Date html generated:
2016_05_18-AM-07_53_14
Last ObjectModification:
2015_12_28-AM-01_07_06
Theory : reals
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