Theorems · Theorem · general topology

liminf_sub_const

∀ {ι : Type u_1} {R : Type u_4} [inst : ConditionallyCompleteLinearOrder R] [inst_1 : TopologicalSpace R]
  [OrderTopology R] (F : Filter ι) [F.NeBot] [inst_4 : AddCommSemigroup R] [inst_5 : Sub R] [ContinuousSub R]
  [OrderedSub R] (f : ι → R) (c : R),
  Filter.IsCoboundedUnder (fun x1 x2 => x1 ≥ x2) F f →
    Filter.IsBoundedUnder (fun x1 x2 => x1 ≥ x2) F f → Filter.liminf (fun i => f i - c) F = Filter.liminf f F - c

liminf (xᵢ - c) = (liminf xᵢ) - c.

Defined in

Mathlib.Topology.Algebra.Order.LiminfLimsup

Cited by

0 results in Mathlib

Foundations

Depth 90 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ConditionallyCompleteLinearOrderTopologicalSpaceOrderTopologyFilter.NeBotAddCommSemigroupSubContinuousSubOrderedSub