Nuprl Lemma : rnexp-converges-ext
∀x:ℝ. ((|x| < r1) ⇒ lim n→∞.x^n = r0)
Proof
Definitions occuring in Statement :
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,
natural_number: $n
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
so_apply: x[s1;s2],
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2;s3;s4],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
uall: ∀[x:A]. B[x],
rationals-dense-ext,
rnexp-converges,
reg-seq-mul: reg-seq-mul(x;y),
bnot: ¬bb,
le_int: i ≤z j,
canonical-bound: canonical-bound(r),
imax: imax(a;b),
reg-seq-adjust: reg-seq-adjust(n;x),
reg-seq-inv: reg-seq-inv(x),
accelerate: accelerate(k;f),
eq_int: (i =z j),
bfalse: ff,
it: ⋅,
btrue: tt,
lt_int: i <z j,
ifthenelse: if b then t else f fi ,
mu-ge: mu-ge(f;n),
rinv: rinv(x),
rmul: a * b,
rdiv: (x/y),
rabs: |x|,
member: t ∈ T
Lemmas referenced :
strict4-spread,
lifting-strict-callbyvalue,
rnexp-converges,
rationals-dense-ext
Rules used in proof :
independent_isectElimination,
voidEquality,
voidElimination,
isect_memberEquality,
baseClosed,
isectElimination,
equalitySymmetry,
equalityTransitivity,
sqequalHypSubstitution,
thin,
sqequalRule,
hypothesis,
extract_by_obid,
instantiate,
cut,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
introduction
Latex:
\mforall{}x:\mBbbR{}. ((|x| < r1) {}\mRightarrow{} lim n\mrightarrow{}\minfty{}.x\^{}n = r0)
Date html generated:
2018_05_22-PM-01_51_16
Last ObjectModification:
2018_05_21-AM-00_09_47
Theory : reals
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