Nuprl Lemma : comparison-test
∀y:ℕ ⟶ ℝ. (Σn.y[n]↓ ⇒ (∀x:ℕ ⟶ ℝ. Σn.x[n]↓ supposing ∀n:ℕ. (|x[n]| ≤ y[n])))
Proof
Definitions occuring in Statement :
series-converges: Σn.x[n]↓,
rleq: x ≤ y,
rabs: |x|,
real: ℝ,
nat: ℕ,
uimplies: b supposing a,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
less_than': less_than'(a;b),
nat: ℕ,
so_lambda: λ2x.t[x],
converges: x[n]↓ as n→∞,
series-sum: Σn.x[n] = a,
series-converges: Σn.x[n]↓,
prop: ℙ,
real: ℝ,
subtype_rel: A ⊆r B,
so_apply: x[s],
uall: ∀[x:A]. B[x],
false: False,
not: ¬A,
and: P ∧ Q,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
all: ∀x:A. B[x],
cauchy: cauchy(n.x[n]),
all-large: ∀large(n).P[n],
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T},
top: Top,
rneq: x ≠ y
Latex:
\mforall{}y:\mBbbN{} {}\mrightarrow{} \mBbbR{}. (\mSigma{}n.y[n]\mdownarrow{} {}\mRightarrow{} (\mforall{}x:\mBbbN{} {}\mrightarrow{} \mBbbR{}. \mSigma{}n.x[n]\mdownarrow{} supposing \mforall{}n:\mBbbN{}. (|x[n]| \mleq{} y[n])))
Date html generated:
2020_05_20-AM-11_20_37
Last ObjectModification:
2020_03_19-PM-00_43_18
Theory : reals
Home
Index