Nuprl Lemma : rleq-limit
∀[x,y:ℕ ⟶ ℝ]. ∀[a,b:ℝ]. (a ≤ b) supposing ((∀n:ℕ. (x[n] ≤ y[n])) and lim n→∞.y[n] = b and lim n→∞.x[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,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
prop: ℙ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T},
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
rsub: x - y
Latex:
\mforall{}[x,y:\mBbbN{} {}\mrightarrow{} \mBbbR{}]. \mforall{}[a,b:\mBbbR{}].
(a \mleq{} b) supposing ((\mforall{}n:\mBbbN{}. (x[n] \mleq{} y[n])) and lim n\mrightarrow{}\minfty{}.y[n] = b and lim n\mrightarrow{}\minfty{}.x[n] = a)
Date html generated:
2020_05_20-AM-11_16_05
Last ObjectModification:
2020_01_02-PM-02_11_36
Theory : reals
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