Nuprl Lemma : near-fixpoint-on-0-1
∀f:[r0, r1] ⟶ℝ
((∀x:ℝ. ((x ∈ [r0, r1]) ⇒ (f[x] ∈ [r0, r1])))
⇒ f[x] continuous for x ∈ [r0, r1]
⇒ (∀e:ℝ. ((r0 < e) ⇒ (∃x:ℝ. ((x ∈ [r0, r1]) ∧ (|f[x] - x| < e))))))
Proof
Definitions occuring in Statement :
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
rccint: [l, u],
i-member: r ∈ I,
rless: x < y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_apply: x[s],
rfun: I ⟶ℝ,
and: P ∧ Q,
cand: A c∧ B,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
or: P ∨ Q,
so_lambda: λ2x.t[x],
label: ...$L... t,
uiff: uiff(P;Q),
rccint: [l, u],
i-member: r ∈ I,
exists: ∃x:A. B[x],
req_int_terms: t1 ≡ t2,
rev_uimplies: rev_uimplies(P;Q),
r-ap: f(x),
guard: {T},
sq_stable: SqStable(P),
squash: ↓T,
rge: x ≥ y,
subtype_rel: A ⊆r B,
true: True
Latex:
\mforall{}f:[r0, r1] {}\mrightarrow{}\mBbbR{}
((\mforall{}x:\mBbbR{}. ((x \mmember{} [r0, r1]) {}\mRightarrow{} (f[x] \mmember{} [r0, r1])))
{}\mRightarrow{} f[x] continuous for x \mmember{} [r0, r1]
{}\mRightarrow{} (\mforall{}e:\mBbbR{}. ((r0 < e) {}\mRightarrow{} (\mexists{}x:\mBbbR{}. ((x \mmember{} [r0, r1]) \mwedge{} (|f[x] - x| < e))))))
Date html generated:
2020_05_20-PM-00_28_38
Last ObjectModification:
2020_01_03-PM-03_50_03
Theory : reals
Home
Index