Nuprl Lemma : fun-converges-to-rexp
lim n→∞.Σ{(x^i)/(i)! | 0≤i≤n} = λx.e^x for x ∈ (-∞, ∞)
Proof
Definitions occuring in Statement :
fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I,
riiint: (-∞, ∞),
rexp: e^x,
rsum: Σ{x[k] | n≤k≤m},
rnexp: x^k1,
int-rdiv: (a)/k1,
fact: (n)!,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
nat_plus: ℕ+,
so_apply: x[s],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
int_upper: {i...},
sq_stable: SqStable(P),
squash: ↓T,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
rless: x < y,
sq_exists: ∃x:A [B[x]],
real: ℝ,
cand: A c∧ B,
rge: x ≥ y,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
so_apply: x[s1;s2],
top: Top,
lelt: i ≤ j < k,
int_seg: {i..j-},
less_than': less_than'(a;b),
le: A ≤ B,
rfun: I ⟶ℝ,
so_lambda: λ2x y.t[x; y],
pointwise-req: x[k] = y[k] for k ∈ [n,m]
Latex:
lim n\mrightarrow{}\minfty{}.\mSigma{}\{(x\^{}i)/(i)! | 0\mleq{}i\mleq{}n\} = \mlambda{}x.e\^{}x for x \mmember{} (-\minfty{}, \minfty{})
Date html generated:
2020_05_20-PM-01_08_32
Last ObjectModification:
2019_12_14-PM-02_55_13
Theory : reals
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