Nuprl Lemma : rinv-converges-to-0
lim n→∞.(r1/r(n + 1)) = r0
Proof
Definitions occuring in Statement : 
converges-to: lim n→∞.x[n] = y
, 
rdiv: (x/y)
, 
int-to-real: r(n)
, 
add: n + m
, 
natural_number: $n
Definitions unfolded in proof : 
converges-to: lim n→∞.x[n] = y
, 
all: ∀x:A. B[x]
, 
member: t ∈ T
, 
sq_exists: ∃x:{A| B[x]}
, 
subtype_rel: A ⊆r B
, 
implies: P 
⇒ Q
, 
prop: ℙ
, 
uall: ∀[x:A]. B[x]
, 
nat_plus: ℕ+
, 
nat: ℕ
, 
so_lambda: λ2x.t[x]
, 
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
, 
so_apply: x[s]
, 
rdiv: (x/y)
, 
itermConstant: "const"
, 
req_int_terms: t1 ≡ t2
, 
uiff: uiff(P;Q)
, 
rev_uimplies: rev_uimplies(P;Q)
, 
le: A ≤ B
, 
subtract: n - m
, 
less_than': less_than'(a;b)
, 
true: True
Lemmas referenced : 
nat_plus_wf, 
nat_plus_subtype_nat, 
le_wf, 
nat_wf, 
all_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, 
itermAdd_wf, 
itermVar_wf, 
intformle_wf, 
int_formula_prop_and_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_less_lemma, 
int_term_value_constant_lemma, 
int_term_value_add_lemma, 
int_term_value_var_lemma, 
int_formula_prop_le_lemma, 
int_formula_prop_wf, 
rless_wf, 
rmul_wf, 
rinv_wf2, 
uiff_transitivity, 
rleq_functionality, 
rabs_functionality, 
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, 
rinv-as-rdiv, 
rleq-int-fractions2, 
false_wf, 
not-lt-2, 
less-iff-le, 
condition-implies-le, 
minus-add, 
minus-one-mul, 
zero-add, 
minus-one-mul-top, 
add-commutes, 
add_functionality_wrt_le, 
add-associates, 
add-zero, 
add-swap, 
le-add-cancel, 
less_than_wf, 
decidable__le, 
int_term_value_mul_lemma, 
rleq-int-fractions, 
rabs-of-nonneg, 
req_weakening
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
cut, 
introduction, 
extract_by_obid, 
hypothesis, 
dependent_set_memberFormation, 
hypothesisEquality, 
applyEquality, 
sqequalHypSubstitution, 
sqequalRule, 
isectElimination, 
thin, 
setElimination, 
rename, 
lambdaEquality, 
functionEquality, 
because_Cache, 
natural_numberEquality, 
addEquality, 
independent_isectElimination, 
inrFormation, 
dependent_functionElimination, 
productElimination, 
independent_functionElimination, 
unionElimination, 
dependent_pairFormation, 
int_eqEquality, 
intEquality, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
independent_pairFormation, 
computeAll, 
dependent_set_memberEquality, 
minusEquality, 
multiplyEquality
Latex:
lim  n\mrightarrow{}\minfty{}.(r1/r(n  +  1))  =  r0
Date html generated:
2017_10_03-AM-08_51_45
Last ObjectModification:
2017_07_28-AM-07_34_49
Theory : reals
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