Nuprl Lemma : fun-converges-iff-cauchy
∀I:Interval. ∀f:ℕ ⟶ I ⟶ℝ. (λn.f[n;x]↓ for x ∈ I) ⇐⇒ λn.f[n;x] is cauchy for x ∈ I)
Proof
Definitions occuring in Statement :
fun-converges: λn.f[n; x]↓ for x ∈ I),
fun-cauchy: λn.f[n; x] is cauchy for x ∈ I,
rfun: I ⟶ℝ,
interval: Interval,
nat: ℕ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x y.t[x; y],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
prop: ℙ,
rev_implies: P ⇐ Q,
fun-cauchy: λn.f[n; x] is cauchy for x ∈ I,
fun-converges: λn.f[n; x]↓ for x ∈ I),
exists: ∃x:A. B[x],
fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
nat: ℕ,
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s],
less_than: a < b,
squash: ↓T,
cand: A c∧ B,
sq_stable: SqStable(P),
guard: {T},
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
uiff: uiff(P;Q),
icompact: icompact(I),
i-nonvoid: i-nonvoid(I),
cauchy: cauchy(n.x[n]),
sq_exists: ∃x:A [B[x]],
ge: i ≥ j ,
req_int_terms: t1 ≡ t2,
converges: x[n]↓ as n→∞,
pi1: fst(t),
converges-to: lim n→∞.x[n] = y
Latex:
\mforall{}I:Interval. \mforall{}f:\mBbbN{} {}\mrightarrow{} I {}\mrightarrow{}\mBbbR{}. (\mlambda{}n.f[n;x]\mdownarrow{} for x \mmember{} I) \mLeftarrow{}{}\mRightarrow{} \mlambda{}n.f[n;x] is cauchy for x \mmember{} I)
Date html generated:
2020_05_20-PM-01_05_44
Last ObjectModification:
2020_01_08-PM-03_16_32
Theory : reals
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