exists_norm_mul_lt — Mathlib

English

Given S ≤ M, m ∈ M, and ε > 0, there exists s ∈ S with ∥m s∥ < ∥mk' S m∥ + ε.

Русский

Для любого подгруппы S и элемента m ∈ M существует s ∈ S такое, что ∥m s∥ < ∥mk' S m∥ + ε.

LaTeX

$$$\exists s \in S, \; \|m s\| < \|mk' S m\| + \varepsilon$$$

Lean4

/-- For any `m : M` and any `0 < ε`, there is `s ∈ S` such that `‖m * s‖ < ‖mk' S m‖ + ε`. -/
@[to_additive /-- For any `m : M` and any `0 < ε`, there is `s ∈ S` such that `‖m + s‖ < ‖mk' S m‖ + ε`. -/
    ]
theorem exists_norm_mul_lt (S : Subgroup M) (m : M) {ε : ℝ} (hε : 0 < ε) : ∃ s ∈ S, ‖m * s‖ < ‖mk' S m‖ + ε :=
  by
  obtain ⟨n : M, hn, hn'⟩ := exists_norm_mk_lt (QuotientGroup.mk' S m) hε
  exact ⟨m⁻¹ * n, by simpa [eq_comm, QuotientGroup.eq] using hn, by simpa⟩

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