Nuprl Lemma : rpowers-converge-ext
∀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 :
member: t ∈ T,
rabs: |x|,
int-to-real: r(n),
rsub: x - y,
rminus: -(x),
canonical-bound: canonical-bound(r),
rpowers-converge,
rnexp-converges,
integer-bound,
rless_transitivity2,
rless-int,
rleq_weakening_rless,
rationals-dense-ext,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
rpowers-converge,
lifting-strict-callbyvalue,
istype-void,
strict4-spread,
rnexp-converges,
integer-bound,
rless_transitivity2,
rless-int,
rleq_weakening_rless,
rationals-dense-ext
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
isect_memberEquality_alt,
voidElimination,
independent_isectElimination
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:
2019_10_29-AM-10_10_35
Last ObjectModification:
2019_04_01-PM-10_59_29
Theory : reals
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