Nuprl Lemma : connectedness-main-lemma-ext
∀x:ℝ. ∀g:ℕ ⟶ ℝ. (lim n→∞.g n = x ⇒ (∀P:ℝ ⟶ 𝔹. ∃z:{z:ℝ| P z = P accelerate(3;x)} . (∃n:{ℕ| (z = (g n))})))
Proof
Definitions occuring in Statement :
converges-to: lim n→∞.x[n] = y,
req: x = y,
accelerate: accelerate(k;f),
real: ℝ,
nat: ℕ,
bool: 𝔹,
all: ∀x:A. B[x],
sq_exists: ∃x:{A| B[x]},
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
connectedness-main-lemma,
weak-continuity-principle-real-ext,
let: let
Lemmas referenced :
connectedness-main-lemma,
weak-continuity-principle-real-ext
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}x:\mBbbR{}. \mforall{}g:\mBbbN{} {}\mrightarrow{} \mBbbR{}.
(lim n\mrightarrow{}\minfty{}.g n = x {}\mRightarrow{} (\mforall{}P:\mBbbR{} {}\mrightarrow{} \mBbbB{}. \mexists{}z:\{z:\mBbbR{}| P z = P accelerate(3;x)\} . (\mexists{}n:\{\mBbbN{}| (z = (g n))\})))
Date html generated:
2017_10_03-AM-10_10_53
Last ObjectModification:
2017_09_13-PM-03_27_40
Theory : reals
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