Nuprl Lemma : series-converges-limit-zero
∀x:ℕ ⟶ ℝ. (Σn.x[n]↓ ⇒ lim n→∞.x[n] = r0)
Proof
Definitions occuring in Statement :
series-converges: Σn.x[n]↓,
converges-to: lim n→∞.x[n] = y,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
series-converges: Σn.x[n]↓,
exists: ∃x:A. B[x],
series-sum: Σn.x[n] = a,
converges-to: lim n→∞.x[n] = y,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
and: P ∧ Q,
sq_exists: ∃x:A [B[x]],
nat: ℕ,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
sq_stable: SqStable(P),
rleq: x ≤ y,
rnonneg: rnonneg(x),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
rge: x ≥ y
Latex:
\mforall{}x:\mBbbN{} {}\mrightarrow{} \mBbbR{}. (\mSigma{}n.x[n]\mdownarrow{} {}\mRightarrow{} lim n\mrightarrow{}\minfty{}.x[n] = r0)
Date html generated:
2020_05_20-AM-11_20_16
Last ObjectModification:
2019_12_14-PM-04_53_55
Theory : reals
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