Nuprl Lemma : rinv-limit
∀x:ℕ ⟶ ℝ. ∀a:ℝ. (lim n→∞.x[n] = a ⇒ (∀n:ℕ. x[n] ≠ r0) ⇒ a ≠ r0 ⇒ lim n→∞.(r1/x[n]) = (r1/a))
Proof
Definitions occuring in Statement :
converges-to: lim n→∞.x[n] = y,
rdiv: (x/y),
rneq: x ≠ y,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n
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],
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
rsub: x - y,
rge: x ≥ y,
guard: {T},
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
nat_plus: ℕ+,
rless: x < y,
sq_exists: ∃x:A [B[x]],
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
rdiv: (x/y),
sq-all-large: ∀large(n).{P[n]},
nat: ℕ,
rat_term_to_real: rat_term_to_real(f;t),
rtermDivide: num "/" denom,
rat_term_ind: rat_term_ind,
rtermVar: rtermVar(var),
rtermConstant: "const",
pi1: fst(t),
rtermMultiply: left "*" right,
pi2: snd(t),
ge: i ≥ j ,
subtype_rel: A ⊆r B,
absval: |i|,
cand: A c∧ B,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T)
Lemmas referenced :
rneq_wf,
int-to-real_wf,
istype-nat,
converges-to_wf,
real_wf,
radd-preserves-rleq,
rsub_wf,
rabs_wf,
radd_wf,
rminus_wf,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermMinus_wf,
rleq_functionality,
radd_functionality,
req_weakening,
rabs_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_minus_lemma,
rleq_weakening_equal,
rabs-difference-symmetry,
rleq_functionality_wrt_implies,
r-triangle-inequality,
rabs-neq-zero,
rmul_preserves_rless,
rdiv_wf,
rless-int,
rless_wf,
rless-int-fractions2,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
rmul_wf,
rinv_wf2,
itermMultiply_wf,
rless_functionality,
req_transitivity,
rinv-mul-as-rdiv,
real_term_value_mul_lemma,
small-reciprocal-real,
istype-le,
assert-rat-term-eq2,
rtermMultiply_wf,
rtermDivide_wf,
rtermConstant_wf,
rtermVar_wf,
rless_transitivity2,
nat_properties,
intformand_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
radd-preserves-rless,
squash_wf,
true_wf,
radd_comm_eq,
subtype_rel_self,
iff_weakening_equal,
rmul-rinv3,
radd_comm,
nat_plus_wf,
int_term_value_mul_lemma,
rneq_functionality,
rmul-int,
rneq-int,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
less_than_wf,
int_subtype_base,
rmul_functionality,
req_inversion,
rinv_functionality2,
rinv-of-rmul,
sq-all-large-and,
rleq_wf,
rpositive-rless,
rabs-positive,
absval_wf,
rabs-rmul,
rabs-rdiv,
rabs-int,
rmul_preserves_rleq,
rinv-as-rdiv,
rmul-rinv,
rleq_weakening_rless,
rabs-rmul-rleq,
square-nonzero,
req-int-fractions,
nequal_wf,
nat_plus_inc_int_nzero,
decidable__equal_int,
req_functionality,
rmul-rdiv-cancel5,
int_entire_a,
subtype_base_sq,
rminus_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
cut,
universeIsType,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
hypothesis,
sqequalRule,
functionIsType,
applyEquality,
lambdaEquality_alt,
because_Cache,
productElimination,
independent_isectElimination,
dependent_functionElimination,
approximateComputation,
int_eqEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
voidElimination,
independent_functionElimination,
closedConclusion,
inrFormation_alt,
independent_pairFormation,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality_alt,
setElimination,
rename,
unionElimination,
dependent_pairFormation_alt,
inhabitedIsType,
imageElimination,
instantiate,
universeEquality,
multiplyEquality,
equalityIstype,
baseApply,
intEquality,
sqequalBase,
cumulativity
Latex:
\mforall{}x:\mBbbN{} {}\mrightarrow{} \mBbbR{}. \mforall{}a:\mBbbR{}. (lim n\mrightarrow{}\minfty{}.x[n] = a {}\mRightarrow{} (\mforall{}n:\mBbbN{}. x[n] \mneq{} r0) {}\mRightarrow{} a \mneq{} r0 {}\mRightarrow{} lim n\mrightarrow{}\minfty{}.(r1/x[n]) = (r1/a))
Date html generated:
2019_10_29-AM-10_22_43
Last ObjectModification:
2019_04_02-PM-04_08_58
Theory : reals
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