English
For any nonzero element a in a monoid with zero and no zero-divisors, the powered term a^n cannot be zero; more precisely, a^n ≠ 0 for all n.
Русский
Для любого ненулевого элемента a в моноиде с нулём и без делителей нуля, степени a^n не равно нулю; то есть для любого n имеем a^n ≠ 0.
LaTeX
$$$a \neq 0 \Rightarrow a^n \neq 0 \quad (n \in \mathbb{N})$$$
Lean4
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence. -/
theorem uniform_geometric_approx {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n :=
by
rw [← hasFPowerSeriesWithinOnBall_univ] at hf
simpa using hf.uniform_geometric_approx h
