Nuprl Lemma : rpowers-converge
∀x:ℝ. (((|x| < r1) ⇒ lim n→∞.x^n = r0) ∧ ((r1 < x) ⇒ lim n →∞.x^n = ∞))
Proof
Definitions occuring in Statement :
converges-to-infinity: lim n →∞.x[n] = ∞,
converges-to: lim n→∞.x[n] = y,
rless: x < y,
rabs: |x|,
rnexp: x^k1,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
rdiv: (x/y),
all-large: ∀large(n).P[n],
converges-to-infinity: lim n →∞.x[n] = ∞,
nat_plus: ℕ+,
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
sq_stable: SqStable(P),
real: ℝ,
subtype_rel: A ⊆r B,
sq_exists: ∃x:{A| B[x]},
rless: x < y,
nat: ℕ,
so_apply: x[s],
so_lambda: λ2x.t[x],
ge: i ≥ j ,
subtract: n - m,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y
Lemmas referenced :
rnexp-converges,
rless_wf,
rabs_wf,
int-to-real_wf,
real_wf,
rdiv_wf,
rsub_wf,
rless-int,
rmul_preserves_rless,
radd_wf,
rmul_wf,
rminus_wf,
rinv_wf2,
rless-implies-rless,
real_term_polynomial,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
itermAdd_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
req-iff-rsub-is-0,
itermMultiply_wf,
real_term_value_mul_lemma,
rless_functionality,
req_transitivity,
itermMinus_wf,
real_term_value_minus_lemma,
radd_functionality,
rmul_functionality,
rmul-rinv,
req_weakening,
rmul-identity1,
rminus_functionality,
squash_wf,
true_wf,
rminus-int,
rleq-int,
rmul_preserves_rleq2,
rleq_weakening_rless,
less_than'_wf,
radd-preserves-rleq,
subtract_wf,
radd-preserves-rless,
rless_transitivity1,
rdiv_functionality,
rsub-int,
rmul-rdiv-cancel2,
radd-int,
add-commutes,
add-swap,
add-associates,
zero-add,
nat_properties,
nat_wf,
all_wf,
rleq_wf,
rnexp_wf,
rabs-bounds,
rleq_transitivity,
rless_transitivity2,
nat_plus_properties,
sq_stable__less_than,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
decidable__lt,
less_than_wf,
integer-bound,
nat_plus_wf,
rpower-greater-one
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
isectElimination,
natural_numberEquality,
independent_pairFormation,
dependent_pairFormation,
independent_isectElimination,
sqequalRule,
inrFormation,
because_Cache,
productElimination,
imageMemberEquality,
baseClosed,
productEquality,
minusEquality,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
lemma_by_obid,
rename,
setElimination,
unionElimination,
addEquality,
dependent_set_memberEquality,
functionEquality,
promote_hyp,
axiomEquality,
independent_pairEquality,
isect_memberFormation
Latex:
\mforall{}x:\mBbbR{}. (((|x| < r1) {}\mRightarrow{} lim n\mrightarrow{}\minfty{}.x\^{}n = r0) \mwedge{} ((r1 < x) {}\mRightarrow{} lim n \mrightarrow{}\minfty{}.x\^{}n = \minfty{}))
Date html generated:
2017_10_03-AM-08_54_50
Last ObjectModification:
2017_07_28-AM-07_36_45
Theory : reals
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