Nuprl Lemma : subsequence-converges
∀a:ℝ. ∀x,y:ℕ ⟶ ℝ.
((∃N:ℕ. ∀n:ℕ. ∃m:ℕ. ((n ≤ m) ∧ (y[n] = x[m])) supposing N ≤ n) ⇒ lim n→∞.x[n] = a ⇒ lim n→∞.y[n] = a)
Proof
Definitions occuring in Statement :
converges-to: lim n→∞.x[n] = y,
req: x = y,
real: ℝ,
nat: ℕ,
uimplies: b supposing a,
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
converges-to: lim n→∞.x[n] = y,
member: t ∈ T,
sq_exists: ∃x:{A| B[x]},
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
prop: ℙ,
nat: ℕ,
so_apply: x[s],
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
exists: ∃x:A. B[x],
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
sq_stable: SqStable(P),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
subtype_rel: A ⊆r B,
squash: ↓T,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
sq_stable__all,
nat_wf,
le_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_properties,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
sq_stable__rleq,
less_than'_wf,
squash_wf,
nat_plus_wf,
converges-to_wf,
exists_wf,
all_wf,
isect_wf,
req_wf,
real_wf,
imax_wf,
imax_nat,
decidable__le,
intformle_wf,
intformeq_wf,
int_formula_prop_le_lemma,
int_formula_prop_eq_lemma,
equal_wf,
imax_lb,
rleq_functionality,
rabs_functionality,
rsub_functionality,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
introduction,
extract_by_obid,
isectElimination,
sqequalRule,
lambdaEquality,
functionEquality,
because_Cache,
applyEquality,
functionExtensionality,
natural_numberEquality,
independent_isectElimination,
inrFormation,
productElimination,
independent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_pairEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
imageElimination,
productEquality,
dependent_set_memberFormation,
dependent_set_memberEquality,
applyLambdaEquality
Latex:
\mforall{}a:\mBbbR{}. \mforall{}x,y:\mBbbN{} {}\mrightarrow{} \mBbbR{}.
((\mexists{}N:\mBbbN{}. \mforall{}n:\mBbbN{}. \mexists{}m:\mBbbN{}. ((n \mleq{} m) \mwedge{} (y[n] = x[m])) supposing N \mleq{} n)
{}\mRightarrow{} lim n\mrightarrow{}\minfty{}.x[n] = a
{}\mRightarrow{} lim n\mrightarrow{}\minfty{}.y[n] = a)
Date html generated:
2017_10_03-AM-08_54_11
Last ObjectModification:
2017_07_28-AM-07_36_12
Theory : reals
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