Theorems · Theorem · general topology
Real.isLUB_of_tendsto_monotoneOn_bddAbove_nat_Ici
∀ {f : ℕ → ℝ} {k : ℕ} {x : ℝ},
Filter.Tendsto f Filter.atTop (nhds x) →
MonotoneOn f (Set.Ici k) → BddAbove (f '' Set.Ici k) → IsLUB (f '' Set.Ici k) xThe limit of an antitone, bounded below sequence f : ℕ → ℝ on Ici k is a least
upper bound of the sequence.
- 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
Around this declaration
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Cites14
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
- Set.imagestatement and proof · cited by 5,609
- nhdsstatement and proof · cited by 5,554
- Filter.Tendstostatement and proof · cited by 3,814
- Filter.atTopstatement and proof · cited by 2,405
- Set.Icistatement and proof · cited by 1,070
- BddAbovestatement and proof · cited by 620
- MonotoneOnstatement and proof · cited by 311
- IsLUBstatement and proof · cited by 280
- tendsto_nhds_uniqueproof · cited by 118
- isLUB_csSupproof · cited by 34
- Set.image_nonemptyproof · cited by 29
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