Theorems · Theorem · general topology
Real.tendsto_atTop_csSup_of_monotoneOn_bddAbove_nat_Ici
∀ {f : ℕ → ℝ} {k : ℕ},
MonotoneOn f (Set.Ici k) → BddAbove (f '' Set.Ici k) → Filter.Tendsto f Filter.atTop (nhds (sSup (f '' Set.Ici k)))A monotone, bounded above sequence f : ℕ → ℝ on Ici k has the finite
limit sSup (f '' Ici k).
- Defined in
- Mathlib.Topology.Instances.Real.Lemmas
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 117 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setproof · cited by 53,352
- 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
- SupSet.sSupstatement and proof · cited by 954
- BddAbovestatement and proof · cited by 620
- MonotoneOnstatement and proof · cited by 311
- Filter.tendsto_add_atTop_iff_natproof · cited by 23
- sSup_rangeproof · cited by 20
Cited by1
Results whose statement or proof uses this declaration.
- Real.isLUB_of_tendsto_monotoneOn_bddAbove_nat_Iciproof · cited by 0