Theorems · Theorem · field theory

Filter.Tendsto.atBot_mul_pos

∀ {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} {C : 𝕜},
  0 < C →
    Filter.Tendsto f l Filter.atBot → Filter.Tendsto g l (nhds C) → Filter.Tendsto (fun x => f x * g x) l Filter.atBot

In a linearly ordered field with the order topology, if f tends to Filter.atBot and g tends to a positive constant C then f * g tends to Filter.atBot.

Defined in

Mathlib.Topology.Algebra.Order.Field

Cited by

2 results in Mathlib

Foundations

Depth 73 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldLinearOrderIsStrictOrderedRingTopologicalSpaceOrderTopology