isLittleO_const_id_cobounded — Mathlib

English

If for every i in a multiset s we have f i = o_l g i and there exists an i ∈ s with f i = o_l g i, then the product over s of f i is o_l the product over s of g i.

Русский

Если для каждого i в мультисете s f_i = o_l g_i и есть i ∈ s с f_i = o_l g_i, то произведение по i∈s f_i = o_l произведения по i∈s g_i.

LaTeX

$$$ (\\forall i \\in s,\\; f i =o[l] g i) \\land (\\Exists i \\in s,\\; f i =o[l] g i) \\Rightarrow (\\lambda x. (s.map (f \\cdot x)).prod) =o[l] (\\lambda x. (s.map (g \\cdot x)).prod) $$$

Lean4

theorem isLittleO_const_id_cobounded (c : F'') : (fun _ => c) =o[Bornology.cobounded E''] id :=
  isLittleO_const_left.2 <| .inr tendsto_norm_cobounded_atTop

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