Nuprl Lemma : fun-comparison-test
∀I:Interval. ∀f,g:ℕ ⟶ I ⟶ℝ.
(Σn.g[n;x]↓ for x ∈ I ⇒ (∀n:ℕ. ∀x:{x:ℝ| x ∈ I} . (|f[n;x]| ≤ g[n;x])) ⇒ Σn.f[n;x]↓ for x ∈ I)
Proof
Definitions occuring in Statement :
fun-series-converges: Σn.f[n; x]↓ for x ∈ I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
rleq: x ≤ y,
rabs: |x|,
real: ℝ,
nat: ℕ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
fun-series-converges: Σn.f[n; x]↓ for x ∈ I,
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
rfun: I ⟶ℝ,
uall: ∀[x:A]. B[x],
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
fun-cauchy: λn.f[n; x] is cauchy for x ∈ I,
label: ...$L... t,
all-large: ∀large(n).P[n],
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
int_upper: {i...},
sq_stable: SqStable(P),
top: Top,
true: True,
less_than': less_than'(a;b),
subtract: n - m,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
req_int_terms: t1 ≡ t2,
rdiv: (x/y)
Latex:
\mforall{}I:Interval. \mforall{}f,g:\mBbbN{} {}\mrightarrow{} I {}\mrightarrow{}\mBbbR{}.
(\mSigma{}n.g[n;x]\mdownarrow{} for x \mmember{} I {}\mRightarrow{} (\mforall{}n:\mBbbN{}. \mforall{}x:\{x:\mBbbR{}| x \mmember{} I\} . (|f[n;x]| \mleq{} g[n;x])) {}\mRightarrow{} \mSigma{}n.f[n;x]\mdownarrow{} for x \mmember{} I)
Date html generated:
2020_05_20-PM-01_06_13
Last ObjectModification:
2020_01_08-AM-10_41_44
Theory : reals
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