Nuprl Lemma : converges-iff-cauchy
∀x:ℕ ⟶ ℝ. (x[n]↓ as n→∞ ⇐⇒ cauchy(n.x[n]))
Proof
Definitions occuring in Statement :
cauchy: cauchy(n.x[n]),
converges: x[n]↓ as n→∞,
real: ℝ,
nat: ℕ,
so_apply: x[s],
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.t[x],
so_apply: x[s],
prop: ℙ,
rev_implies: P ⇐ Q,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
sq_stable: SqStable(P),
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
decidable: Dec(P),
ge: i ≥ j ,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
uimplies: b supposing a,
nat: ℕ,
sq_exists: ∃x:A [B[x]],
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
nat_plus: ℕ+,
cauchy: cauchy(n.x[n]),
converges-to: lim n→∞.x[n] = y,
exists: ∃x:A. B[x],
converges: x[n]↓ as n→∞,
uiff: uiff(P;Q),
rge: x ≥ y,
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
subtype_rel: A ⊆r B,
rdiv: (x/y),
real: ℝ,
rational-approx: (x within 1/n),
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
sq_type: SQType(T),
reg-seq-add: reg-seq-add(x;y),
rabs: |x|,
rminus: -(x),
rleq2: rleq2(x;y),
rmax: rmax(x;y),
int_upper: {i...},
reg-seq-mul: reg-seq-mul(x;y),
int-to-real: r(n),
regular-int-seq: k-regular-seq(f),
subtract: n - m,
absval: |i|
Latex:
\mforall{}x:\mBbbN{} {}\mrightarrow{} \mBbbR{}. (x[n]\mdownarrow{} as n\mrightarrow{}\minfty{} \mLeftarrow{}{}\mRightarrow{} cauchy(n.x[n]))
Date html generated:
2020_05_20-AM-11_08_47
Last ObjectModification:
2019_12_28-PM-11_47_44
Theory : reals
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