Nuprl Lemma : common-limit-squeeze
∀a,b,c:ℕ ⟶ ℝ.
((∀n:ℕ. ((a[n] ≤ a[n + 1]) ∧ (a[n + 1] ≤ b[n + 1]) ∧ (b[n + 1] ≤ b[n])))
⇒ lim n→∞.c[n] = r0
⇒ (∀n:ℕ. r0≤b[n] - a[n]≤c[n])
⇒ (∃y:ℝ. (lim n→∞.a[n] = y ∧ lim n→∞.b[n] = y)))
Proof
Definitions occuring in Statement :
converges-to: lim n→∞.x[n] = y,
rbetween: x≤y≤z,
rleq: x ≤ y,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
and: P ∧ Q,
nat: ℕ,
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,
converges: x[n]↓ as n→∞,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
converges-to: lim n→∞.x[n] = y,
cauchy: cauchy(n.x[n]),
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
rbetween: x≤y≤z,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
subtype_rel: A ⊆r B,
sq_type: SQType(T),
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
cand: A c∧ B,
sq-all-large: ∀large(n).{P[n]}
Latex:
\mforall{}a,b,c:\mBbbN{} {}\mrightarrow{} \mBbbR{}.
((\mforall{}n:\mBbbN{}. ((a[n] \mleq{} a[n + 1]) \mwedge{} (a[n + 1] \mleq{} b[n + 1]) \mwedge{} (b[n + 1] \mleq{} b[n])))
{}\mRightarrow{} lim n\mrightarrow{}\minfty{}.c[n] = r0
{}\mRightarrow{} (\mforall{}n:\mBbbN{}. r0\mleq{}b[n] - a[n]\mleq{}c[n])
{}\mRightarrow{} (\mexists{}y:\mBbbR{}. (lim n\mrightarrow{}\minfty{}.a[n] = y \mwedge{} lim n\mrightarrow{}\minfty{}.b[n] = y)))
Date html generated:
2020_05_20-AM-11_09_49
Last ObjectModification:
2019_12_14-PM-00_54_16
Theory : reals
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