English
The rational numbers are densely embedded in the p-adic numbers: for every x ∈ ℚ_p and every ε>0, there exists r ∈ ℚ with |x - r|_p < ε.
Русский
Рациональные числа плотно встроены в p-адические числа: для каждого x ∈ ℚ_p и ε>0 существует r ∈ ℚ SUCH that |x - r|_p < ε.
LaTeX
$$$\forall x \in \mathbb{Q}_p\;\forall \varepsilon>0:\; \exists r \in \mathbb{Q}: |x - r|_p < \varepsilon$$$
Lean4
theorem rat_dense (q : ℚ_[p]) {ε : ℝ} (hε : 0 < ε) : ∃ r : ℚ, ‖q - r‖ < ε :=
let ⟨ε', hε'l, hε'r⟩ := exists_rat_btwn hε
let ⟨r, hr⟩ := rat_dense' q (ε := ε') (by simpa using hε'l)
⟨r, lt_trans (by simpa [Norm.norm] using hr) hε'r⟩