Theorems · Theorem · field theory

bdd_le_mul_tendsto_zero

∀ {𝕜 : Type u_1} {α : Type u_2} [inst : Field 𝕜] [inst_1 : LinearOrder 𝕜] [IsStrictOrderedRing 𝕜]
  [inst_3 : TopologicalSpace 𝕜] [OrderTopology 𝕜] {l : Filter α} {f g : α → 𝕜} {b B : 𝕜},
  (∀ᶠ (x : α) in l, b ≤ f x) →
    (∀ᶠ (x : α) in l, f x ≤ B) → Filter.Tendsto g l (nhds 0) → Filter.Tendsto (fun x => f x * g x) l (nhds 0)

If g tends to zero and there exist constants b B : 𝕜 such that eventually b ≤ f x| ≤ B, then the product f * g tends to zero.

Defined in

Mathlib.Topology.Algebra.Order.Field

Cited by

1 results in Mathlib

Foundations

Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldLinearOrderIsStrictOrderedRingTopologicalSpaceOrderTopology