tendsto_intCast_atBot_sup_atTop_cobounded

English

For a NormedRing α, the map n ↦ n (as a natural) under the natural cast tends to coboundedness atTop, i.e., tends to infinity in the bornology sense.

Русский

Для нормированного кольца α отображение $n$ через естественное включение стремится к бесконечности в смысле ковбounded.

LaTeX

$$$$\\\\ Tendsto(\\\\text{Nat.cast}) atTop \\\\to (\\\\text{Bornology}.cobounded α) $$$$

Lean4

theorem tendsto_intCast_atBot_sup_atTop_cobounded [NormedRing α] [NormSMulClass ℤ α] [Nontrivial α] :
    Tendsto Int.cast (atBot ⊔ atTop) (Bornology.cobounded α) :=
  by
  rw [← tendsto_norm_atTop_iff_cobounded]
  simpa [norm_intCast_eq_abs_mul_norm_one] using
    tendsto_intCast_atTop_atTop |>.comp (tendsto_abs_atBot_atTop.sup tendsto_abs_atTop_atTop) |>.atTop_mul_const
      (norm_pos_iff.mpr one_ne_zero)

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