Theorems · Theorem · sequences and series
HasProd.tendsto_prod_nat
∀ {M : Type u_1} [inst : CommMonoid M] [inst_1 : TopologicalSpace M] {m : M} {f : ℕ → M},
HasProd f m → Filter.Tendsto (fun n => ∏ i ∈ Finset.range n, f i) Filter.atTop (nhds m)If f : ℕ → M has product m, then the partial products ∏ i ∈ range n, f i converge
to m.
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommMonoidTopologicalSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- TopologicalSpacestatement and proof · cited by 24,529
- nhdsstatement · cited by 5,554
- Filter.Tendstostatement · cited by 3,814
- Filter.atTopstatement · cited by 2,405
- Finset.prodstatement · cited by 2,356
- CommMonoidstatement and proof · cited by 2,264
- SummationFilter.unconditionalstatement and proof · cited by 2,068
- Finset.rangestatement · cited by 1,341
- Filter.Tendsto.compproof · cited by 560
- HasProdstatement and proof · cited by 157
- Filter.tendsto_finset_rangeproof · cited by 15
Cited by6
Results whose statement or proof uses this declaration.
- EulerProduct.eulerProduct_completely_multiplicativeproof · cited by 2
- Multipliable.hasProd_iff_tendsto_natproof · cited by 1
- EulerProduct.eulerProductproof · cited by 1
- Multipliable.tendsto_prod_tprod_natproof · cited by 0
- tendsto_prod_nat_addproof · cited by 0
- Multipliable.tprod_le_of_prod_range_leproof · cited by 0