instContinuousAdd — Mathlib

English

Multiplication on ℕ∞ is a continuous binary operation with respect to the order topology.

Русский

Умножение на ℕ∞ является непрерывной бинарной операцией в топологии порядка.

LaTeX

$$$$\\text{Continuous}(\\cdot \\cdot ): \\mathbb{N}_{\\infty} \\times \\mathbb{N}_{\\infty} \\to \\mathbb{N}_{\\infty}$$$$

Lean4

instance : ContinuousAdd ℕ∞ := by
  refine ⟨continuous_iff_continuousAt.2 fun (a, b) ↦ ?_⟩
  match a, b with
  | ⊤, _ => exact tendsto_nhds_top_mono' continuousAt_fst fun p ↦ le_add_right le_rfl
  | (a : ℕ), ⊤ => exact tendsto_nhds_top_mono' continuousAt_snd fun p ↦ le_add_left le_rfl
  | (a : ℕ), (b : ℕ) => simp [ContinuousAt, nhds_prod_eq, tendsto_pure_nhds]

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