/-- Assume that a function `f` has a derivative at every point of a set `s`. Then one may cover `s`
with countably many closed sets `t n` on which `f` is well approximated by linear maps `A n`. -/
theorem exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt [SecondCountableTopology F]
    (f : E → F) (s : Set E) (f' : E → E →L[ℝ] F) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
    (r : (E →L[ℝ] F) → ℝ≥0) (rpos : ∀ A, r A ≠ 0) :
    ∃ (t : ℕ → Set E) (A : ℕ → E →L[ℝ] F),
      (∀ n, IsClosed (t n)) ∧
        (s ⊆ ⋃ n, t n) ∧
          (∀ n, ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧
            (s.Nonempty → ∀ n, ∃ y ∈ s, A n = f' y) := by
  /- Choose countably many linear maps `f' z`. For every such map, if `f` has a derivative at `x`
    close enough to `f' z`, then `f y - f x` is well approximated by `f' z (y - x)` for `y` close
    enough to `x`, say on a ball of radius `r` (or even `u n` for some `n`, where `u` is a fixed
    sequence tending to `0`).
    Let `M n z` be the points where this happens. Then this set is relatively closed inside `s`,
    and moreover in every closed ball of radius `u n / 3` inside it the map is well approximated by
    `f' z`. Using countably many closed balls to split `M n z` into small diameter subsets
    `K n z p`, one obtains the desired sets `t q` after reindexing.
    -/
  -- exclude the trivial case where `s` is empty
  rcases eq_empty_or_nonempty s with (rfl | hs)
  · refine' ⟨fun _ => ∅, fun _ => 0, _, _, _, _⟩ <;> simp
  -- we will use countably many linear maps. Select these from all the derivatives since the
  -- space of linear maps is second-countable
  obtain ⟨T, T_count, hT⟩ :
    ∃ T : Set s,
      T.Countable ∧ ⋃ x ∈ T, ball (f' (x : E)) (r (f' x)) = ⋃ x : s, ball (f' x) (r (f' x)) :=
    TopologicalSpace.isOpen_iUnion_countable _ fun x => isOpen_ball
  -- fix a sequence `u` of positive reals tending to zero.
  obtain ⟨u, _, u_pos, u_lim⟩ :
    ∃ u : ℕ → ℝ, StrictAnti u ∧ (∀ n : ℕ, 0 < u n) ∧ Tendsto u atTop (𝓝 0) :=
    exists_seq_strictAnti_tendsto (0 : ℝ)
  -- `M n z` is the set of points `x` such that `f y - f x` is close to `f' z (y - x)` for `y`
  -- in the ball of radius `u n` around `x`.
  let M : ℕ → T → Set E := fun n z =>
    {x | x ∈ s ∧ ∀ y ∈ s ∩ ball x (u n), ‖f y - f x - f' z (y - x)‖ ≤ r (f' z) * ‖y - x‖}
  -- As `f` is differentiable everywhere on `s`, the sets `M n z` cover `s` by design.
  have s_subset : ∀ x ∈ s, ∃ (n : ℕ) (z : T), x ∈ M n z := by
    intro x xs
    obtain ⟨z, zT, hz⟩ : ∃ z ∈ T, f' x ∈ ball (f' (z : E)) (r (f' z)) := by
      have : f' x ∈ ⋃ z ∈ T, ball (f' (z : E)) (r (f' z)) := by
        rw [hT]
        refine' mem_iUnion.2 ⟨⟨x, xs⟩, _⟩
        simpa only [mem_ball, Subtype.coe_mk, dist_self] using (rpos (f' x)).bot_lt
      rwa [mem_iUnion₂, bex_def] at this
    obtain ⟨ε, εpos, hε⟩ : ∃ ε : ℝ, 0 < ε ∧ ‖f' x - f' z‖ + ε ≤ r (f' z) := by
      refine' ⟨r (f' z) - ‖f' x - f' z‖, _, le_of_eq (by abel)⟩
      simpa only [sub_pos] using mem_ball_iff_norm.mp hz
    obtain ⟨δ, δpos, hδ⟩ :
      ∃ (δ : ℝ), 0 < δ ∧ ball x δ ∩ s ⊆ {y | ‖f y - f x - (f' x) (y - x)‖ ≤ ε * ‖y - x‖} :=
      Metric.mem_nhdsWithin_iff.1 ((hf' x xs).isLittleO.def εpos)
    obtain ⟨n, hn⟩ : ∃ n, u n < δ := ((tendsto_order.1 u_lim).2 _ δpos).exists
    refine' ⟨n, ⟨z, zT⟩, ⟨xs, _⟩⟩
    intro y hy
    calc
      ‖f y - f x - (f' z) (y - x)‖ = ‖f y - f x - (f' x) (y - x) + (f' x - f' z) (y - x)‖ := by
        congr 1
        simp only [ContinuousLinearMap.coe_sub', map_sub, Pi.sub_apply]
        abel
      _ ≤ ‖f y - f x - (f' x) (y - x)‖ + ‖(f' x - f' z) (y - x)‖ := (norm_add_le _ _)
      _ ≤ ε * ‖y - x‖ + ‖f' x - f' z‖ * ‖y - x‖ := by
        refine' add_le_add (hδ _) (ContinuousLinearMap.le_op_norm _ _)
        rw [inter_comm]
        exact inter_subset_inter_right _ (ball_subset_ball hn.le) hy
      _ ≤ r (f' z) * ‖y - x‖ := by
        rw [← add_mul, add_comm]
        exact mul_le_mul_of_nonneg_right hε (norm_nonneg _)
  -- the sets `M n z` are relatively closed in `s`, as all the conditions defining it are clearly
  -- closed
  have closure_M_subset : ∀ n z, s ∩ closure (M n z) ⊆ M n z := by
    rintro n z x ⟨xs, hx⟩
    refine' ⟨xs, fun y hy => _⟩
    obtain ⟨a, aM, a_lim⟩ : ∃ a : ℕ → E, (∀ k, a k ∈ M n z) ∧ Tendsto a atTop (𝓝 x) :=
      mem_closure_iff_seq_limit.1 hx
    have L1 :
      Tendsto (fun k : ℕ => ‖f y - f (a k) - (f' z) (y - a k)‖) atTop
        (𝓝 ‖f y - f x - (f' z) (y - x)‖) := by
      apply Tendsto.norm
      have L : Tendsto (fun k => f (a k)) atTop (𝓝 (f x)) := by
        apply (hf' x xs).continuousWithinAt.tendsto.comp
        apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ a_lim
        exact eventually_of_forall fun k => (aM k).1
      apply Tendsto.sub (tendsto_const_nhds.sub L)
      exact ((f' z).continuous.tendsto _).comp (tendsto_const_nhds.sub a_lim)
    have L2 : Tendsto (fun k : ℕ => (r (f' z) : ℝ) * ‖y - a k‖) atTop (𝓝 (r (f' z) * ‖y - x‖)) :=
      (tendsto_const_nhds.sub a_lim).norm.const_mul _
    have I : ∀ᶠ k in atTop, ‖f y - f (a k) - (f' z) (y - a k)‖ ≤ r (f' z) * ‖y - a k‖ := by
      have L : Tendsto (fun k => dist y (a k)) atTop (𝓝 (dist y x)) :=
        tendsto_const_nhds.dist a_lim
      filter_upwards [(tendsto_order.1 L).2 _ hy.2]
      intro k hk
      exact (aM k).2 y ⟨hy.1, hk⟩
    exact le_of_tendsto_of_tendsto L1 L2 I
  -- choose a dense sequence `d p`
  rcases TopologicalSpace.exists_dense_seq E with ⟨d, hd⟩
  -- split `M n z` into subsets `K n z p` of small diameters by intersecting with the ball
  -- `closedBall (d p) (u n / 3)`.
  let K : ℕ → T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)
  -- on the sets `K n z p`, the map `f` is well approximated by `f' z` by design.
  have K_approx : ∀ (n) (z : T) (p), ApproximatesLinearOn f (f' z) (s ∩ K n z p) (r (f' z)) := by
    intro n z p x hx y hy
    have yM : y ∈ M n z := closure_M_subset _ _ ⟨hy.1, hy.2.1⟩
    refine' yM.2 _ ⟨hx.1, _⟩
    calc
      dist x y ≤ dist x (d p) + dist y (d p) := dist_triangle_right _ _ _
      _ ≤ u n / 3 + u n / 3 := (add_le_add hx.2.2 hy.2.2)
      _ < u n := by linarith [u_pos n]
  -- the sets `K n z p` are also closed, again by design.
  have K_closed : ∀ (n) (z : T) (p), IsClosed (K n z p) := fun n z p =>
    isClosed_closure.inter isClosed_ball
  -- reindex the sets `K n z p`, to let them only depend on an integer parameter `q`.
  obtain ⟨F, hF⟩ : ∃ F : ℕ → ℕ × T × ℕ, Function.Surjective F := by
    haveI : Encodable T := T_count.toEncodable
    have : Nonempty T := by
      rcases eq_empty_or_nonempty T with (rfl | hT)
      · rcases hs with ⟨x, xs⟩
        rcases s_subset x xs with ⟨n, z, _⟩
        exact False.elim z.2
      · exact hT.coe_sort
    inhabit ↥T
    exact ⟨_, Encodable.surjective_decode_iget (ℕ × T × ℕ)⟩
  -- these sets `t q = K n z p` will do
  refine'
    ⟨fun q => K (F q).1 (F q).2.1 (F q).2.2, fun q => f' (F q).2.1, fun n => K_closed _ _ _,
      fun x xs => _, fun q => K_approx _ _ _, fun _ q => ⟨(F q).2.1, (F q).2.1.1.2, rfl⟩⟩
  -- the only fact that needs further checking is that they cover `s`.
  -- we already know that any point `x ∈ s` belongs to a set `M n z`.
  obtain ⟨n, z, hnz⟩ : ∃ (n : ℕ) (z : T), x ∈ M n z := s_subset x xs
  -- by density, it also belongs to a ball `closedBall (d p) (u n / 3)`.
  obtain ⟨p, hp⟩ : ∃ p : ℕ, x ∈ closedBall (d p) (u n / 3) := by
    have : Set.Nonempty (ball x (u n / 3)) := by simp only [nonempty_ball]; linarith [u_pos n]
    obtain ⟨p, hp⟩ : ∃ p : ℕ, d p ∈ ball x (u n / 3) := hd.exists_mem_open isOpen_ball this
    

Proof state:
case intro
E : Type u_1
F✝ : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : NormedAddCommGroup F✝
inst✝¹ : NormedSpace ℝ F✝
s✝ : Set E
f✝ : E → E
f'✝ : E → E →L[ℝ] E
inst✝ : SecondCountableTopology F✝
f : E → F✝
s : Set E
f' : E → E →L[ℝ] F✝
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
r : (E →L[ℝ] F✝) → ℝ≥0
rpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0
hs : Set.Nonempty s
T : Set ↑s
T_count : Set.Countable T
hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x))
u : ℕ → ℝ
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
M : ℕ → ↑T → Set E :=
  fun n z => {x | x ∈ s ∧ ∀ y ∈ s ∩ ball x (u n), ‖f y - f x - (f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}
s_subset : ∀ x ∈ s, ∃ n z, x ∈ M n z
closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z
d : ℕ → E
hd : DenseRange d
K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)
K_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))
K_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)
F : ℕ → ℕ × ↑T × ℕ
hF : Function.Surjective F
x : E
xs : x ∈ s
n : ℕ
z : ↑T
hnz : x ∈ M n z
this : Set.Nonempty (ball x (u n / 3))
p : ℕ
hp : d p ∈ ball x (u n / 3)
⊢ ∃ p, x ∈ closedBall (d p) (u n / 3)

Next tactic:
exact ⟨p, (mem_ball'.1 hp).le⟩