/-- Existence of minimizers
Let `u` be a point in a real inner product space, and let `K` be a nonempty complete convex subset.
Then there exists a (unique) `v` in `K` that minimizes the distance `‖u - v‖` to `u`.
 -/
theorem exists_norm_eq_iInf_of_complete_convex {K : Set F} (ne : K.Nonempty) (h₁ : IsComplete K)
    (h₂ : Convex ℝ K) : ∀ u : F, ∃ v ∈ K, ‖u - v‖ = ⨅ w : K, ‖u - w‖ := fun u => by
  let δ := ⨅ w : K, ‖u - w‖
  letI : Nonempty K := ne.to_subtype
  have zero_le_δ : 0 ≤ δ := le_ciInf fun _ => norm_nonneg _
  have δ_le : ∀ w : K, δ ≤ ‖u - w‖ := ciInf_le ⟨0, Set.forall_range_iff.2 fun _ => norm_nonneg _⟩
  have δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖ := fun w hw => δ_le ⟨w, hw⟩
  -- Step 1: since `δ` is the infimum, can find a sequence `w : ℕ → K` in `K`
  -- such that `‖u - w n‖ < δ + 1 / (n + 1)` (which implies `‖u - w n‖ --> δ`);
  -- maybe this should be a separate lemma
  have exists_seq : ∃ w : ℕ → K, ∀ n, ‖u - w n‖ < δ + 1 / (n + 1) := by
    have hδ : ∀ n : ℕ, δ < δ + 1 / (n + 1) := fun n =>
      lt_add_of_le_of_pos le_rfl Nat.one_div_pos_of_nat
    have h := fun n => exists_lt_of_ciInf_lt (hδ n)
    let w : ℕ → K := fun n => Classical.choose (h n)
    exact ⟨w, fun n => Classical.choose_spec (h n)⟩
  rcases exists_seq with ⟨w, hw⟩
  have norm_tendsto : Tendsto (fun n => ‖u - w n‖) atTop (nhds δ) := by
    have h : Tendsto (fun _ : ℕ => δ) atTop (nhds δ) := tendsto_const_nhds
    have h' : Tendsto (fun n : ℕ => δ + 1 / (n + 1)) atTop (nhds δ) := by
      convert h.add tendsto_one_div_add_atTop_nhds_0_nat
      simp only [add_zero]
    exact tendsto_of_tendsto_of_tendsto_of_le_of_le h h' (fun x => δ_le _) fun x => le_of_lt (hw _)
  -- Step 2: Prove that the sequence `w : ℕ → K` is a Cauchy sequence
  have seq_is_cauchy : CauchySeq fun n => (w n : F) := by
    rw [cauchySeq_iff_le_tendsto_0]
    -- splits into three goals
    let b := fun n : ℕ => 8 * δ * (1 / (n + 1)) + 4 * (1 / (n + 1)) * (1 / (n + 1))
    use fun n => sqrt (b n)
    constructor
    -- first goal :  `∀ (n : ℕ), 0 ≤ sqrt (b n)`
    intro n
    exact sqrt_nonneg _
    constructor
    -- second goal : `∀ (n m N : ℕ), N ≤ n → N ≤ m → dist ↑(w n) ↑(w m) ≤ sqrt (b N)`
    intro p q N hp hq
    let wp := (w p : F)
    let wq := (w q : F)
    let a := u - wq
    let b := u - wp
    let half := 1 / (2 : ℝ)
    let div := 1 / ((N : ℝ) + 1)
    have :
      4 * ‖u - half • (wq + wp)‖ * ‖u - half • (wq + wp)‖ + ‖wp - wq‖ * ‖wp - wq‖ =
        2 * (‖a‖ * ‖a‖ + ‖b‖ * ‖b‖) :=
      calc
        4 * ‖u - half • (wq + wp)‖ * ‖u - half • (wq + wp)‖ + ‖wp - wq‖ * ‖wp - wq‖ =
            2 * ‖u - half • (wq + wp)‖ * (2 * ‖u - half • (wq + wp)‖) + ‖wp - wq‖ * ‖wp - wq‖ :=
          by ring
        _ =
            absR (2 : ℝ) * ‖u - half • (wq + wp)‖ * (absR (2 : ℝ) * ‖u - half • (wq + wp)‖) +
              ‖wp - wq‖ * ‖wp - wq‖ := by
          rw [_root_.abs_of_nonneg]
          exact zero_le_two
        _ =
            ‖(2 : ℝ) • (u - half • (wq + wp))‖ * ‖(2 : ℝ) • (u - half • (wq + wp))‖ +
              ‖wp - wq‖ * ‖wp - wq‖ :=
          by simp [norm_smul]
        _ = ‖a + b‖ * ‖a + b‖ + ‖a - b‖ * ‖a - b‖ := by
          rw [smul_sub, smul_smul, mul_one_div_cancel (_root_.two_ne_zero : (2 : ℝ) ≠ 0), ←
            one_add_one_eq_two, add_smul]
          simp only [one_smul]
          have eq₁ : wp - wq = a - b := (sub_sub_sub_cancel_left _ _ _).symm
          have eq₂ : u + u - (wq + wp) = a + b
          show u + u - (wq + wp) = u - wq + (u - wp)
          abel
          rw [eq₁, eq₂]
        _ = 2 * (‖a‖ * ‖a‖ + ‖b‖ * ‖b‖) := parallelogram_law_with_norm ℝ _ _
    have eq : δ ≤ ‖u - half • (wq + wp)‖ := by
      rw [smul_add]
      apply δ_le'
      apply h₂
      repeat' exact Subtype.mem _
      repeat' exact le_of_lt one_half_pos
      exact add_halves 1
    have eq₁ : 4 * δ * δ ≤ 4 * ‖u - half • (wq + wp)‖ * ‖u - half • (wq + wp)‖ := by
      simp_rw [mul_assoc]
      gcongr
    have eq₂ : ‖a‖ ≤ δ + div :=
        le_trans (le_of_lt <| hw q) (add_le_add_left (Nat.one_div_le_one_div hq) _)
    have eq₂' : ‖b‖ ≤ δ + div :=
        le_trans (le_of_lt <| hw p) (add_le_add_left (Nat.one_div_le_one_div hp) _)
    rw [dist_eq_norm]
    apply nonneg_le_nonneg_of_sq_le_sq
    · exact sqrt_nonneg _
    rw [mul_self_sqrt]
    calc
      ‖wp - wq‖ * ‖wp - wq‖ =
          2 * (‖a‖ * ‖a‖ + ‖b‖ * ‖b‖) - 4 * ‖u - half • (wq + wp)‖ * ‖u - half • (wq + wp)‖ := by
        simp [← this]
      _ ≤ 2 * (‖a‖ * ‖a‖ + ‖b‖ * ‖b‖) - 4 * δ * δ := by gcongr
      _ ≤ 2 * ((δ + div) * (δ + div) + (δ + div) * (δ + div)) - 4 * δ * δ := by gcongr
      _ = 8 * δ * div + 4 * div * div := by ring
    

Proof state:
case h.right.left.h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : InnerProductSpace ℝ F
K : Set F
ne : Set.Nonempty K
h₁ : IsComplete K
h₂ : Convex ℝ K
u : F
δ : ℝ := ⨅ w, ‖u - ↑w‖
this✝ : Nonempty ↑K := Set.Nonempty.to_subtype ne
zero_le_δ : 0 ≤ δ
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
w : ℕ → ↑K
hw : ∀ (n : ℕ), ‖u - ↑(w n)‖ < δ + 1 / (↑n + 1)
norm_tendsto : Tendsto (fun n => ‖u - ↑(w n)‖) atTop (𝓝 δ)
b✝ : ℕ → ℝ := fun n => 8 * δ * (1 / (↑n + 1)) + 4 * (1 / (↑n + 1)) * (1 / (↑n + 1))
p q N : ℕ
hp : N ≤ p
hq : N ≤ q
wp : F := ↑(w p)
wq : F := ↑(w q)
a : F := u - wq
b : F := u - wp
half : ℝ := 1 / 2
div : ℝ := 1 / (↑N + 1)
this : 4 * ‖u - half • (wq + wp)‖ * ‖u - half • (wq + wp)‖ + ‖wp - wq‖ * ‖wp - wq‖ = 2 * (‖a‖ * ‖a‖ + ‖b‖ * ‖b‖)
eq : δ ≤ ‖u - half • (wq + wp)‖
eq₁ : 4 * δ * δ ≤ 4 * ‖u - half • (wq + wp)‖ * ‖u - half • (wq + wp)‖
eq₂ : ‖a‖ ≤ δ + div
eq₂' : ‖b‖ ≤ δ + div
⊢ 0 ≤ b✝ N
case h.right.right
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : InnerProductSpace ℝ F
K : Set F
ne : Set.Nonempty K
h₁ : IsComplete K
h₂ : Convex ℝ K
u : F
δ : ℝ := ⨅ w, ‖u - ↑w‖
this : Nonempty ↑K := Set.Nonempty.to_subtype ne
zero_le_δ : 0 ≤ δ
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
w : ℕ → ↑K
hw : ∀ (n : ℕ), ‖u - ↑(w n)‖ < δ + 1 / (↑n + 1)
norm_tendsto : Tendsto (fun n => ‖u - ↑(w n)‖) atTop (𝓝 δ)
b : ℕ → ℝ := fun n => 8 * δ * (1 / (↑n + 1)) + 4 * (1 / (↑n + 1)) * (1 / (↑n + 1))
⊢ Tendsto (fun n => sqrt (b n)) atTop (𝓝 0)

Next tactic:
positivity