tendsto_prod_tprod_nat — Mathlib

English

Equality of automorphizations with respect to another equivalence, linking quotient and action representations.

Русский

Равенство автоматизаций при разных представлениях фактор-группы и действия.

LaTeX

$$$\\operatorname{QuotientGroup}.automorphize f = \\operatorname{MulAction}.automorphize f$$$

Lean4

/-- If `f : ℕ → M` is multipliable, then the partial products `∏ i ∈ range n, f i` converge
to `∏' i, f i`. -/
@[to_additive /-- If `f : ℕ → M` is summable, then the partial sums `∑ i ∈ range n, f i` converge
    to `∑' i, f i`. -/
    ]
theorem tendsto_prod_tprod_nat {f : ℕ → M} (h : Multipliable f) :
    Tendsto (fun n ↦ ∏ i ∈ range n, f i) atTop (𝓝 (∏' i, f i)) :=
  h.hasProd.tendsto_prod_nat

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