Nuprl Lemma : rational-approx-converges-to
∀[x:ℝ]. lim n→∞.(x within 1/n + 1) = x
Proof
Definitions occuring in Statement :
converges-to: lim n→∞.x[n] = y,
rational-approx: (x within 1/n),
real: ℝ,
uall: ∀[x:A]. B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
converges-to: lim n→∞.x[n] = y,
all: ∀x:A. B[x],
sq_exists: ∃x:{A| B[x]},
member: t ∈ T,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
prop: ℙ,
nat_plus: ℕ+,
nat: ℕ,
so_lambda: λ2x.t[x],
real: ℝ,
le: A ≤ B,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
top: Top,
less_than': less_than'(a;b),
true: True,
rneq: x ≠ y,
guard: {T},
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y
Lemmas referenced :
nat_plus_subtype_nat,
le_wf,
nat_wf,
all_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rational-approx_wf,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_properties,
nat_plus_properties,
full-omega-unsat,
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,
nat_plus_wf,
real_wf,
less-iff-le,
add-swap,
itermAdd_wf,
intformle_wf,
int_term_value_add_lemma,
int_formula_prop_le_lemma,
rleq-int-fractions,
decidable__le,
itermMultiply_wf,
int_term_value_mul_lemma,
rleq_functionality,
rabs-difference-symmetry,
req_weakening,
rleq_functionality_wrt_implies,
rational-approx-property,
rleq_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
dependent_set_memberFormation,
cut,
hypothesisEquality,
applyEquality,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
sqequalRule,
isectElimination,
thin,
setElimination,
rename,
lambdaEquality,
functionEquality,
because_Cache,
dependent_set_memberEquality,
addEquality,
natural_numberEquality,
productElimination,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
isect_memberEquality,
voidEquality,
intEquality,
minusEquality,
inrFormation,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
multiplyEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[x:\mBbbR{}]. lim n\mrightarrow{}\minfty{}.(x within 1/n + 1) = x
Date html generated:
2017_10_03-AM-08_51_34
Last ObjectModification:
2017_06_30-PM-04_10_52
Theory : reals
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