Nuprl Lemma : simple-converges-to
∀x:ℕ ⟶ ℝ. ∀a,c:ℝ. ((∀n:ℕ. (|(x n) - a| ≤ ((r1/r(2^n)) * c))) ⇒ lim n→∞.x n = a)
Proof
Definitions occuring in Statement :
converges-to: lim n→∞.x[n] = y,
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
exp: i^n,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
sq_type: SQType(T),
false: False,
prop: ℙ,
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
uiff: uiff(P;Q),
int_upper: {i...},
le: A ≤ B,
rneq: x ≠ y,
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
rge: x ≥ y,
converges-to: lim n→∞.x[n] = y
Lemmas referenced :
istype-nat,
rleq_wf,
rabs_wf,
rsub_wf,
rmul_wf,
rdiv_wf,
int-to-real_wf,
exp_wf2,
rneq-int,
not_functionality_wrt_implies,
equal-wf-base,
rationals_wf,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
equal_functionality_wrt_subtype_rel2,
int-subtype-rationals,
int_nzero-rational,
exp_wf3,
subtype_base_sq,
nequal_wf,
real_wf,
r-archimedean,
nat_plus_wf,
decidable__equal_int,
nat_plus_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
le_weakening2,
mul-commutes,
zero-mul,
exp-positive-stronger,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-less_than,
log-property,
log_wf,
add_nat_wf,
add-is-int-iff,
intformand_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
false_wf,
exp-nondecreasing,
itermMultiply_wf,
int_term_value_mul_lemma,
rless-int,
rless_wf,
rmul_preserves_rleq,
rinv_wf2,
itermSubtract_wf,
rleq_functionality,
req_transitivity,
rmul-rinv3,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
rleq-int,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rmul-int,
rmul_functionality,
req_weakening,
rmul-rinv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalRule,
functionIsType,
introduction,
extract_by_obid,
hypothesis,
universeIsType,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
hypothesisEquality,
closedConclusion,
natural_numberEquality,
independent_isectElimination,
dependent_functionElimination,
productElimination,
independent_functionElimination,
baseApply,
baseClosed,
intEquality,
lambdaEquality_alt,
because_Cache,
dependent_set_memberEquality_alt,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
voidElimination,
equalityIstype,
sqequalBase,
inhabitedIsType,
rename,
setElimination,
unionElimination,
dependent_set_memberFormation_alt,
approximateComputation,
dependent_pairFormation_alt,
isect_memberEquality_alt,
multiplyEquality,
independent_pairFormation,
imageMemberEquality,
addEquality,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
int_eqEquality,
inrFormation_alt
Latex:
\mforall{}x:\mBbbN{} {}\mrightarrow{} \mBbbR{}. \mforall{}a,c:\mBbbR{}. ((\mforall{}n:\mBbbN{}. (|(x n) - a| \mleq{} ((r1/r(2\^{}n)) * c))) {}\mRightarrow{} lim n\mrightarrow{}\minfty{}.x n = a)
Date html generated:
2019_10_29-AM-10_10_51
Last ObjectModification:
2019_02_11-PM-02_08_29
Theory : reals
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