/-- If `δ` is small enough, a `(1-δ)`-separated set in the ball of radius `2` also has cardinality
at most `multiplicity E`. -/
theorem exists_goodδ :
    ∃ δ : ℝ, 0 < δ ∧ δ < 1 ∧ ∀ s : Finset E, (∀ c ∈ s, ‖c‖ ≤ 2) →
      (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) → s.card ≤ multiplicity E := by
  

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
⊢ ∃ δ,
    0 < δ ∧
      δ < 1 ∧
        ∀ (s : Finset E),
          (∀ c ∈ s, ‖c‖ ≤ 2) → (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) → Finset.card s ≤ multiplicity E

Next tactic:
classical
  /- This follows from a compactness argument: otherwise, one could extract a converging
    subsequence, to obtain a `1`-separated set in the ball of radius `2` with cardinality
    `N = multiplicity E + 1`. To formalize this, we work with functions `Fin N → E`.
     -/
  by_contra! h
  set N := multiplicity E + 1 with hN
  have :
    ∀ δ : ℝ, 0 < δ → ∃ f : Fin N → E, (∀ i : Fin N, ‖f i‖ ≤ 2) ∧
      ∀ i j, i ≠ j → 1 - δ ≤ ‖f i - f j‖ := by
    intro δ hδ
    rcases lt_or_le δ 1 with (hδ' | hδ')
    · rcases h δ hδ hδ' with ⟨s, hs, h's, s_card⟩
      obtain ⟨f, f_inj, hfs⟩ : ∃ f : Fin N → E, Function.Injective f ∧ range f ⊆ ↑s := by
        have : Fintype.card (Fin N) ≤ s.card := by simp only [Fintype.card_fin]; exact s_card
        rcases Function.Embedding.exists_of_card_le_finset this with ⟨f, hf⟩
        exact ⟨f, f.injective, hf⟩
      simp only [range_subset_iff, Finset.mem_coe] at hfs
      refine' ⟨f, fun i => hs _ (hfs i), fun i j hij => h's _ (hfs i) _ (hfs j) (f_inj.ne hij)⟩
    · exact
        ⟨fun _ => 0, fun i => by simp; norm_num, fun i j _ => by
          simpa only [norm_zero, sub_nonpos, sub_self]⟩
  -- For `δ > 0`, `F δ` is a function from `fin N` to the ball of radius `2` for which two points
  -- in the image are separated by `1 - δ`.
  choose! F hF using this
  -- Choose a converging subsequence when `δ → 0`.
  have : ∃ f : Fin N → E, (∀ i : Fin N, ‖f i‖ ≤ 2) ∧ ∀ i j, i ≠ j → 1 ≤ ‖f i - f j‖ := by
    obtain ⟨u, _, zero_lt_u, hu⟩ :
      ∃ u : ℕ → ℝ,
        (∀ m n : ℕ, m < n → u n < u m) ∧ (∀ n : ℕ, 0 < u n) ∧ Filter.Tendsto u Filter.atTop (𝓝 0) :=
      exists_seq_strictAnti_tendsto (0 : ℝ)
    have A : ∀ n, F (u n) ∈ closedBall (0 : Fin N → E) 2 := by
      intro n
      simp only [pi_norm_le_iff_of_nonneg zero_le_two, mem_closedBall, dist_zero_right,
        (hF (u n) (zero_lt_u n)).left, forall_const]
    obtain ⟨f, fmem, φ, φ_mono, hf⟩ :
      ∃ f ∈ closedBall (0 : Fin N → E) 2,
        ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto ((F ∘ u) ∘ φ) atTop (𝓝 f) :=
      IsCompact.tendsto_subseq (isCompact_closedBall _ _) A
    refine' ⟨f, fun i => _, fun i j hij => _⟩
    · simp only [pi_norm_le_iff_of_nonneg zero_le_two, mem_closedBall, dist_zero_right] at fmem
      exact fmem i
    · have A : Tendsto (fun n => ‖F (u (φ n)) i - F (u (φ n)) j‖) atTop (𝓝 ‖f i - f j‖) :=
        ((hf.apply i).sub (hf.apply j)).norm
      have B : Tendsto (fun n => 1 - u (φ n)) atTop (𝓝 (1 - 0)) :=
        tendsto_const_nhds.sub (hu.comp φ_mono.tendsto_atTop)
      rw [sub_zero] at B
      exact le_of_tendsto_of_tendsto' B A fun n => (hF (u (φ n)) (zero_lt_u _)).2 i j hij
  rcases this with ⟨f, hf, h'f⟩
  -- the range of `f` contradicts the definition of `multiplicity E`.
  have finj : Function.Injective f := by
    intro i j hij
    by_contra h
    have : 1 ≤ ‖f i - f j‖ := h'f i j h
    simp only [hij, norm_zero, sub_self] at this
    exact lt_irrefl _ (this.trans_lt zero_lt_one)
  let s := Finset.image f Finset.univ
  have s_card : s.card = N := by rw [Finset.card_image_of_injective _ finj]; exact Finset.card_fin N
  have hs : ∀ c ∈ s, ‖c‖ ≤ 2 := by
    simp only [hf, forall_apply_eq_imp_iff, forall_const, forall_exists_index, Finset.mem_univ,
      Finset.mem_image, true_and]
  have h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖ := by
    simp only [forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ, Finset.mem_image,
      Ne.def, exists_true_left, forall_apply_eq_imp_iff, forall_true_left, true_and]
    intro i j hij
    have : i ≠ j := fun h => by rw [h] at hij; exact hij rfl
    exact h'f i j this
  have : s.card ≤ multiplicity E := card_le_multiplicity hs h's
  rw [s_card, hN] at this
  exact lt_irrefl _ ((Nat.lt_succ_self (multiplicity E)).trans_le this)