Nuprl Lemma : const-rmul-limit-with-bound
∀x:ℕ ⟶ ℝ. ∀a,c:ℝ. ∀m:ℕ+. ((|c| ≤ r(m)) ⇒ lim n→∞.x[n] = a ⇒ lim n→∞.c * x[n] = c * a)
Proof
Definitions occuring in Statement :
converges-to: lim n→∞.x[n] = y,
rleq: x ≤ y,
rabs: |x|,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
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,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
nat_plus: ℕ+,
sq_exists: ∃x:{A| B[x]},
nat: 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 ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
cand: A c∧ B,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rless: x < y,
rdiv: (x/y)
Lemmas referenced :
converges-to_wf,
nat_wf,
rleq_wf,
rabs_wf,
int-to-real_wf,
nat_plus_wf,
real_wf,
mul_nat_plus,
le_wf,
all_wf,
rsub_wf,
rmul_wf,
rdiv_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,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
mul_bounds_1b,
zero-rleq-rabs,
rleq-int-fractions2,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
int_term_value_mul_lemma,
uimplies_transitivity,
rleq_functionality_wrt_implies,
rmul_functionality_wrt_rleq2,
rleq_weakening_equal,
uiff_transitivity,
rleq_functionality,
rabs_functionality,
req_weakening,
rabs-rmul,
rmul-is-positive,
rmul_functionality,
rdiv_functionality,
req_inversion,
rmul-int,
rinv_wf2,
req_transitivity,
rinv-of-rmul,
rmul-rinv3,
rinv-as-rdiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
because_Cache,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
hypothesis,
setElimination,
rename,
functionEquality,
dependent_functionElimination,
dependent_set_memberEquality,
independent_functionElimination,
natural_numberEquality,
independent_isectElimination,
inrFormation,
productElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
multiplyEquality,
inlFormation,
productEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}x:\mBbbN{} {}\mrightarrow{} \mBbbR{}. \mforall{}a,c:\mBbbR{}. \mforall{}m:\mBbbN{}\msupplus{}. ((|c| \mleq{} r(m)) {}\mRightarrow{} lim n\mrightarrow{}\minfty{}.x[n] = a {}\mRightarrow{} lim n\mrightarrow{}\minfty{}.c * x[n] = c * a)
Date html generated:
2017_10_03-AM-09_05_27
Last ObjectModification:
2017_07_28-AM-07_41_36
Theory : reals
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