Nuprl Lemma : subsequence-mconverges-to
∀[X:Type]. ∀[d:metric(X)]. ∀[a:X].
∀x,y:ℕ ⟶ X. (subsequence(a,b.a ≡ b;n.x[n];n.y[n]) ⇒ lim n→∞.x[n] = a ⇒ lim n→∞.y[n] = a)
Proof
Definitions occuring in Statement :
mconverges-to: lim n→∞.x[n] = y,
meq: x ≡ y,
metric: metric(X),
subsequence: subsequence(a,b.E[a; b];m.x[m];n.y[n]),
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
mconverges-to: lim n→∞.x[n] = y,
subsequence: subsequence(a,b.E[a; b];m.x[m];n.y[n]),
member: t ∈ T,
sq_exists: ∃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),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
sq_stable: SqStable(P),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
squash: ↓T,
uiff: uiff(P;Q),
metric: metric(X),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
req_int_terms: t1 ≡ t2
Lemmas referenced :
sq_stable__all,
nat_wf,
le_wf,
rleq_wf,
mdist_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_properties,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-le,
sq_stable__rleq,
le_witness_for_triv,
imax_wf,
imax_nat,
decidable__le,
intformle_wf,
intformeq_wf,
int_formula_prop_le_lemma,
int_formula_prop_eq_lemma,
imax_lb,
nat_plus_wf,
mconverges-to_wf,
istype-nat,
subsequence_wf,
meq_wf,
metric_wf,
istype-universe,
meq-same,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq_weakening,
itermSubtract_wf,
req-iff-rsub-is-0,
mdist_functionality,
rleq_functionality,
req_weakening,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
cut,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
introduction,
extract_by_obid,
isectElimination,
sqequalRule,
lambdaEquality_alt,
functionEquality,
because_Cache,
applyEquality,
closedConclusion,
natural_numberEquality,
independent_isectElimination,
inrFormation_alt,
productElimination,
independent_functionElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberFormation_alt,
dependent_set_memberEquality_alt,
applyLambdaEquality,
equalityIstype,
functionIsType,
instantiate,
universeEquality
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[a:X].
\mforall{}x,y:\mBbbN{} {}\mrightarrow{} X. (subsequence(a,b.a \mequiv{} b;n.x[n];n.y[n]) {}\mRightarrow{} lim n\mrightarrow{}\minfty{}.x[n] = a {}\mRightarrow{} lim n\mrightarrow{}\minfty{}.y[n] = a)
Date html generated:
2019_10_30-AM-06_39_07
Last ObjectModification:
2019_10_02-AM-10_52_01
Theory : reals
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