theorem hofer {X : Type*} [MetricSpace X] [CompleteSpace X] (x : X) (ε : ℝ) (ε_pos : 0 < ε)
    {ϕ : X → ℝ} (cont : Continuous ϕ) (nonneg : ∀ y, 0 ≤ ϕ y) : ∃ ε' > 0, ∃ x' : X,
    ε' ≤ ε ∧ d x' x ≤ 2 * ε ∧ ε * ϕ x ≤ ε' * ϕ x' ∧ ∀ y, d x' y ≤ ε' → ϕ y ≤ 2 * ϕ x' := by
  by_contra H
  have reformulation : ∀ (x') (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x' := by
    intro x' k
    rw [div_mul_eq_mul_div, le_div_iff, mul_assoc, mul_le_mul_left ε_pos, mul_comm]
    positivity
  -- Now let's specialize to `ε/2^k`
  replace H :
    ∀ k : ℕ, ∀ x', d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ x' → ∃ y, d x' y ≤ ε / 2 ^ k ∧ 2 * ϕ x' < ϕ y
  · intro k x'
    push_neg at H
    have := H (ε / 2 ^ k) (by simp [ε_pos]) x' (by simp [ε_pos.le, one_le_two])
    simpa [reformulation] using this
  clear reformulation
  haveI : Nonempty X := ⟨x⟩
  choose! F hF using H
  -- Use the axiom of choice
  -- Now define u by induction starting at x, with u_{n+1} = F(n, u_n)
  let u : ℕ → X := fun n => Nat.recOn n x F
  -- The properties of F translate to properties of u
  have hu :
    ∀ n,
      d (u n) x ≤ 2 * ε ∧ 2 ^ n * ϕ x ≤ ϕ (u n) →
        d (u n) (u <| n + 1) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u <| n + 1) := by
    intro n
    exact hF n (u n)
  clear hF
  -- Key properties of u, to be proven by induction
  have key : ∀ n, d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1)) := by
    intro n
    induction' n using Nat.case_strong_induction_on with n IH
    · specialize hu 0
      simpa [mul_nonneg_iff, zero_le_one, ε_pos.le, le_refl] using hu
    have A : d (u (n + 1)) x ≤ 2 * ε := by
      rw [dist_comm]
      let r := range (n + 1) -- range (n+1) = {0, ..., n}
      calc
        d (u 0) (u (n + 1)) ≤ ∑ i in r, d (u i) (u <| i + 1) := dist_le_range_sum_dist u (n + 1)
        _ ≤ ∑ i in r, ε / 2 ^ i :=
          (sum_le_sum fun i i_in => (IH i <| Nat.lt_succ_iff.mp <| Finset.mem_range.mp i_in).1)
        _ = ∑ i in r, (1 / 2) ^ i * ε := by
          congr with i
          field_simp
        _ = (∑ i in r, ((1 : ℝ) / 2) ^ i) * ε := Finset.sum_mul.symm
        _ ≤ 2 * ε := by gcongr; apply sum_geometric_two_le
    have B : 2 ^ (n + 1) * ϕ x ≤ ϕ (u (n + 1)) := by
      refine' @geom_le (ϕ ∘ u) _ zero_le_two (n + 1) fun m hm => _
      exact (IH _ <| Nat.lt_add_one_iff.1 hm).2.le
    exact hu (n + 1) ⟨A, B⟩
  cases' forall_and.mp key with key₁ key₂
  clear hu key
  -- Hence u is Cauchy
  have cauchy_u : CauchySeq u := by
    refine' cauchySeq_of_le_geometric _ ε one_half_lt_one fun n => _
    simpa only [one_div, inv_pow] using key₁ n
  -- So u converges to some y
  obtain ⟨y, limy⟩ : ∃ y, Tendsto u atTop (𝓝 y)
  

Proof state:
X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
this : Nonempty X
F : ℕ → X → X
u : ℕ → X := fun n => Nat.recOn n x F
key₁ : ∀ (x : ℕ), d (u x) (u (x + 1)) ≤ ε / 2 ^ x
key₂ : ∀ (x : ℕ), 2 * ϕ (u x) < ϕ (u (x + 1))
cauchy_u : CauchySeq u
⊢ ∃ y, Tendsto u atTop (𝓝 y)
case intro.intro
X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
this : Nonempty X
F : ℕ → X → X
u : ℕ → X := fun n => Nat.recOn n x F
key₁ : ∀ (x : ℕ), d (u x) (u (x + 1)) ≤ ε / 2 ^ x
key₂ : ∀ (x : ℕ), 2 * ϕ (u x) < ϕ (u (x + 1))
cauchy_u : CauchySeq u
y : X
limy : Tendsto u atTop (𝓝 y)
⊢ False

Next tactic:
exact CompleteSpace.complete cauchy_u