Theorems · Theorem · approximation theory
Filter.Tendsto.cesaro_smul
∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {u : ℕ → E} {l : E},
Filter.Tendsto u Filter.atTop (nhds l) →
Filter.Tendsto (fun n => (↑n)⁻¹ • ∑ i ∈ Finset.range n, u i) Filter.atTop (nhds l)The Cesaro average of a converging sequence converges to the same limit.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 163 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites30
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
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- nhdsstatement and proof · cited by 5,554
- Finset.sumstatement and proof · cited by 5,195
- Filter.Tendstostatement and proof · cited by 3,814
- Filter.atTopstatement and proof · cited by 2,405
- Filter.univ_mem'proof · cited by 1,672
- Filter.mp_memproof · cited by 1,537
- LT.lt.ne'proof · cited by 1,417
- one_smulproof · cited by 1,374
- Finset.rangestatement and proof · cited by 1,341
Cited by1
Results whose statement or proof uses this declaration.
- Filter.Tendsto.cesaroproof · cited by 0