Nuprl Lemma : ratio-test-ext
∀x:ℕ ⟶ ℝ. ∀N:ℕ.
((∀c:{c:ℝ| (r0 ≤ c) ∧ (c < r1)} . ((∀n:{N...}. (|x[n + 1]| ≤ (c * |x[n]|))) ⇒ Σn.x[n]↓))
∧ (∀c:{c:ℝ| r1 < c} . ((∀n:{N...}. ((c * |x[n]|) < |x[n + 1]|)) ⇒ Σn.x[n]↑)))
Proof
Definitions occuring in Statement :
series-diverges: Σn.x[n]↑,
series-converges: Σn.x[n]↓,
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
int_upper: {i...},
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
member: t ∈ T,
so_apply: x[s],
rabs: |x|,
subtract: n - m,
ratio-test,
comparison-test-ext,
series-converges-tail2-ext,
series-converges-rmul-ext,
geometric-series-converges-ext,
series-diverges-trivially,
rless_transitivity2,
rless_functionality,
decidable__le,
any: any x,
rleq_functionality_wrt_implies,
rless-iff4,
rless-iff-large-diff,
rleq_functionality,
regular-less-iff,
decidable__and,
decidable__not,
decidable__less_than',
decidable__implies,
decidable__false,
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.t[x],
top: Top,
uimplies: b supposing a
Lemmas referenced :
ratio-test,
lifting-strict-decide,
istype-void,
strict4-decide,
lifting-strict-less,
comparison-test-ext,
series-converges-tail2-ext,
series-converges-rmul-ext,
geometric-series-converges-ext,
series-diverges-trivially,
rless_transitivity2,
rless_functionality,
decidable__le,
rleq_functionality_wrt_implies,
rless-iff4,
rless-iff-large-diff,
rleq_functionality,
regular-less-iff,
decidable__and,
decidable__not,
decidable__less_than',
decidable__implies,
decidable__false
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:\mBbbN{} {}\mrightarrow{} \mBbbR{}. \mforall{}N:\mBbbN{}.
((\mforall{}c:\{c:\mBbbR{}| (r0 \mleq{} c) \mwedge{} (c < r1)\} . ((\mforall{}n:\{N...\}. (|x[n + 1]| \mleq{} (c * |x[n]|))) {}\mRightarrow{} \mSigma{}n.x[n]\mdownarrow{}))
\mwedge{} (\mforall{}c:\{c:\mBbbR{}| r1 < c\} . ((\mforall{}n:\{N...\}. ((c * |x[n]|) < |x[n + 1]|)) {}\mRightarrow{} \mSigma{}n.x[n]\muparrow{})))
Date html generated:
2019_10_29-AM-10_26_40
Last ObjectModification:
2019_04_02-AM-10_05_04
Theory : reals
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