Nuprl Lemma : radd-limit
∀x,y:ℕ ⟶ ℝ. ∀a,b:ℝ. (lim n→∞.x[n] = a ⇒ lim n→∞.y[n] = b ⇒ lim n→∞.x[n] + y[n] = a + b)
Proof
Definitions occuring in Statement :
converges-to: lim n→∞.x[n] = y,
radd: a + b,
real: ℝ,
nat: ℕ,
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,
converges-to: lim n→∞.x[n] = y,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
prop: ℙ,
sq_exists: ∃x:{A| B[x]},
nat: ℕ,
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s],
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rsub: x - y
Lemmas referenced :
mul_nat_plus,
less_than_wf,
imax_wf,
imax_nat,
nat_wf,
nat_properties,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
equal_wf,
le_wf,
imax_lb,
all_wf,
rleq_wf,
rabs_wf,
rsub_wf,
radd_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
rless_wf,
nat_plus_wf,
converges-to_wf,
real_wf,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
r-triangle-inequality,
rmul_wf,
rminus_wf,
uiff_transitivity,
rleq_functionality,
rabs_functionality,
radd_functionality,
rminus-radd,
req_weakening,
req_inversion,
radd-assoc,
radd_comm,
radd-ac,
rminus-as-rmul,
itermMultiply_wf,
int_term_value_mul_lemma,
rleq-int-fractions,
itermAdd_wf,
int_term_value_add_lemma,
radd_functionality_wrt_rleq,
req_transitivity,
radd-rdiv,
rdiv_functionality,
radd-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
hypothesis,
dependent_functionElimination,
thin,
introduction,
extract_by_obid,
isectElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
hypothesisEquality,
baseClosed,
setElimination,
rename,
dependent_set_memberFormation,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
because_Cache,
productElimination,
functionEquality,
applyEquality,
functionExtensionality,
inrFormation,
minusEquality,
multiplyEquality,
addEquality
Latex:
\mforall{}x,y:\mBbbN{} {}\mrightarrow{} \mBbbR{}. \mforall{}a,b:\mBbbR{}. (lim n\mrightarrow{}\minfty{}.x[n] = a {}\mRightarrow{} lim n\mrightarrow{}\minfty{}.y[n] = b {}\mRightarrow{} lim n\mrightarrow{}\minfty{}.x[n] + y[n] = a + b)
Date html generated:
2017_10_03-AM-09_04_53
Last ObjectModification:
2017_07_28-AM-07_41_12
Theory : reals
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