Nuprl Lemma : connectedness-main-lemma
∀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 :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
prop: ℙ,
false: False,
subtype_rel: A ⊆r B,
real: ℝ,
so_lambda: λ2x.t[x],
so_apply: x[s],
converges-to: lim n→∞.x[n] = y,
sq_exists: ∃x:A [B[x]],
nat: ℕ,
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
sq_stable: SqStable(P),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
squash: ↓T,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
int_seg: {i..j-},
lelt: i ≤ j < k,
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
less_than': less_than'(a;b),
stable: Stable{P}
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:
2020_05_20-PM-00_07_02
Last ObjectModification:
2020_01_08-AM-10_39_01
Theory : reals
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