not_real_isEquiv_padic — Mathlib

English

For κ continuous, (κ(1)−1)^n → 0 as n→∞.

Русский

Пусть κ непрерывна; (κ(1)−1)^n стремится к 0 при n → ∞.

LaTeX

$$Tendsto (κ 1 − 1) ^ n atTop (𝓝 0)$$

Lean4

/-- The standard absolute value on `ℚ` is not equivalent to any `p`-adic absolute value. -/
theorem not_real_isEquiv_padic (p : ℕ) [Fact p.Prime] : ¬real.IsEquiv (padic p) :=
  by
  rw [isEquiv_iff_exists_rpow_eq]
  rintro ⟨c, hc₀, hc⟩
  apply_fun (· 2) at hc
  simp only [real_eq_abs, abs_ofNat, cast_ofNat] at hc
  exact ((padic_le_one p 2).trans_lt <| one_lt_rpow one_lt_two hc₀).ne' hc

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