Nuprl Lemma : geometric-series-converges
∀c:{c:ℝ| (r0 ≤ c) ∧ (c < r1)} . Σn.c^n = (r1/r1 - c)
Proof
Definitions occuring in Statement :
series-sum: Σn.x[n] = a,
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rnexp: x^k1,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
series-sum: Σn.x[n] = a,
converges-to: lim n→∞.x[n] = y,
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
rless: x < y,
sq_exists: ∃x:A [B[x]],
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
sq_stable: SqStable(P),
squash: ↓T,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
nat: ℕ,
ge: i ≥ j ,
subtype_rel: A ⊆r B,
real: ℝ,
so_lambda: λ2x.t[x],
le: A ≤ B,
less_than': less_than'(a;b),
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q),
less_than: a < b,
lelt: i ≤ j < k,
int_seg: {i..j-},
top: Top,
true: True,
pi2: snd(t),
rtermDivide: num "/" denom,
rtermConstant: "const",
rtermMultiply: left "*" right,
rtermSubtract: left "-" right,
pi1: fst(t),
rtermVar: rtermVar(var),
rat_term_ind: rat_term_ind,
rtermMinus: rtermMinus(num),
rat_term_to_real: rat_term_to_real(f;t)
Latex:
\mforall{}c:\{c:\mBbbR{}| (r0 \mleq{} c) \mwedge{} (c < r1)\} . \mSigma{}n.c\^{}n = (r1/r1 - c)
Date html generated:
2020_05_20-AM-11_21_56
Last ObjectModification:
2019_12_15-PM-06_54_22
Theory : reals
Home
Index