Theorems · Theorem · general topology
Real.isGLB_of_tendsto_antitone_bddBelow
∀ {ι : Type u_1} [inst : Preorder ι] [Nonempty ι] [IsDirected ι fun x1 x2 => x1 ≤ x2] {f : ι → ℝ} {x : ℝ},
Filter.Tendsto f Filter.atTop (nhds x) → Antitone f → BddBelow (Set.range f) → IsGLB (Set.range f) xThe limit of an antitone, bounded below function f : ι → ℝ is a greatest lower bound
of the function.
- Defined in
- Mathlib.Topology.Instances.Real.Lemmas
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 153 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- PreorderNonemptyIsDirected
Around this declaration
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Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- Preorderstatement and proof · cited by 7,952
- nhdsstatement and proof · cited by 5,554
- Set.rangestatement and proof · cited by 4,705
- Filter.Tendstostatement and proof · cited by 3,814
- Filter.atTopstatement and proof · cited by 2,405
- Antitonestatement and proof · cited by 563
- BddBelowstatement and proof · cited by 401
- IsGLBstatement and proof · cited by 213
- tendsto_nhds_uniqueproof · cited by 118
- IsDirectedstatement and proof · cited by 16
- tendsto_atTop_ciInfproof · cited by 8
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