Nuprl Lemma : fun-converges-to-rsub
∀I:Interval. ∀f1,f2:ℕ ⟶ I ⟶ℝ. ∀g1,g2:I ⟶ℝ.
(lim n→∞.f1[n;x] = λy.g1[y] for x ∈ I
⇒ lim n→∞.f2[n;x] = λy.g2[y] for x ∈ I
⇒ lim n→∞.f1[n;x] - f2[n;x] = λy.g1[y] - g2[y] for x ∈ I)
Proof
Definitions occuring in Statement :
fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I,
rfun: I ⟶ℝ,
interval: Interval,
rsub: x - y,
nat: ℕ,
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_lambda: λ2x y.t[x; y],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat_plus: ℕ+,
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,
and: P ∧ Q,
guard: {T},
less_than: a < b,
squash: ↓T,
cand: A c∧ B,
sq_stable: SqStable(P),
int_upper: {i...},
subinterval: I ⊆ J ,
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat: ℕ,
ge: i ≥ j ,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y
Latex:
\mforall{}I:Interval. \mforall{}f1,f2:\mBbbN{} {}\mrightarrow{} I {}\mrightarrow{}\mBbbR{}. \mforall{}g1,g2:I {}\mrightarrow{}\mBbbR{}.
(lim n\mrightarrow{}\minfty{}.f1[n;x] = \mlambda{}y.g1[y] for x \mmember{} I
{}\mRightarrow{} lim n\mrightarrow{}\minfty{}.f2[n;x] = \mlambda{}y.g2[y] for x \mmember{} I
{}\mRightarrow{} lim n\mrightarrow{}\minfty{}.f1[n;x] - f2[n;x] = \mlambda{}y.g1[y] - g2[y] for x \mmember{} I)
Date html generated:
2020_05_20-PM-01_05_17
Last ObjectModification:
2019_12_14-PM-02_56_37
Theory : reals
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