Theorems · Theorem · sequences and series

Monotone.cauchySeq_series_mul_of_tendsto_zero_of_bounded

∀ {E : Type u_4} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {b : ℝ} {f : ℕ → ℝ} {z : ℕ → E},
  Monotone f →
    Filter.Tendsto f Filter.atTop (nhds 0) →
      (∀ (n : ℕ), ‖∑ i ∈ Finset.range n, z i‖ ≤ b) → CauchySeq fun n => ∑ i ∈ Finset.range n, f i • z i

Dirichlet's test for monotone sequences.

Defined in

Mathlib.Analysis.SpecificLimits.Normed

Cited by

2 results in Mathlib

Foundations

Depth 162 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpace