tendsto_one_iff_norm_tendsto_zero

English

The NN-norm of a constant function a across the index set is bounded by the NN-norm of a: ∥fun _ : ι => a∥₊ ≤ ∥a∥₊.

Русский

NN-норма константной функции по индексам не превышает NN-норму элемента: ∥fun _ : ι => a∥₊ ≤ ∥a∥₊.

LaTeX

$$$\|\mathrm{const}_a\|_+ \le \|a\|_+$$$

Lean4

@[to_additive]
theorem tendsto_one_iff_norm_tendsto_zero {f : α → E} {a : Filter α} : Tendsto f a (𝓝 1) ↔ Tendsto (‖f ·‖) a (𝓝 0) :=
  tendsto_iff_norm_div_tendsto_zero.trans <| by simp only [div_one]

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