/-- The Bernstein approximations
```
∑ k : Fin (n+1), f (k/n : ℝ) * n.choose k * x^k * (1-x)^(n-k)
```
for a continuous function `f : C([0,1], ℝ)` converge uniformly to `f` as `n` tends to infinity.

This is the proof given in [Richard Beals' *Analysis, an introduction*][beals-analysis], §7D,
and reproduced on wikipedia.
-/
theorem bernsteinApproximation_uniform (f : C(I, ℝ)) :
    Tendsto (fun n : ℕ => bernsteinApproximation n f) atTop (𝓝 f) := by
  simp only [Metric.nhds_basis_ball.tendsto_right_iff, Metric.mem_ball, dist_eq_norm]
  intro ε h
  let δ := δ f ε h
  have nhds_zero := tendsto_const_div_atTop_nhds_0_nat (2 * ‖f‖ * δ ^ (-2 : ℤ))
  filter_upwards [nhds_zero.eventually (gt_mem_nhds (half_pos h)), eventually_gt_atTop 0] with n nh
    npos'
  have npos : 0 < (n : ℝ) := by positivity
  -- As `[0,1]` is compact, it suffices to check the inequality pointwise.
  rw [ContinuousMap.norm_lt_iff _ h]
  intro x
  -- The idea is to split up the sum over `k` into two sets,
  -- `S`, where `x - k/n < δ`, and its complement.
  let S := S f ε h n x
  calc
    |(bernsteinApproximation n f - f) x| = |bernsteinApproximation n f x - f x| := rfl
    _ = |bernsteinApproximation n f x - f x * 1| := by rw [mul_one]
    _ = |bernsteinApproximation n f x - f x * ∑ k : Fin (n + 1), bernstein n k x| := by
      rw [bernstein.probability]
    _ = |∑ k : Fin (n + 1), (f k/ₙ - f x) * bernstein n k x| := by
      simp [bernsteinApproximation, Finset.mul_sum, sub_mul]
    _ ≤ ∑ k : Fin (n + 1), |(f k/ₙ - f x) * bernstein n k x| := (Finset.abs_sum_le_sum_abs _ _)
    _ = ∑ k : Fin (n + 1), |f k/ₙ - f x| * bernstein n k x := by
      simp_rw [abs_mul, abs_eq_self.mpr bernstein_nonneg]
    _ = (∑ k in S, |f k/ₙ - f x| * bernstein n k x) + ∑ k in Sᶜ, |f k/ₙ - f x| * bernstein n k x :=
      (S.sum_add_sum_compl _).symm
    -- We'll now deal with the terms in `S` and the terms in `Sᶜ` in separate calc blocks.
    _ < ε / 2 + ε / 2 :=
      (add_lt_add_of_le_of_lt ?_ ?_)
    _ = ε := add_halves ε
  · -- We now work on the terms in `S`: uniform continuity and `bernstein.probability`
    -- quickly give us a bound.
    calc
      ∑ k in S, |f k/ₙ - f x| * bernstein n k x ≤ ∑ k in S, ε / 2 * bernstein n k x := by
        gcongr with _ m
        exact le_of_lt (lt_of_mem_S m)
      _ = ε / 2 * ∑ k in S, bernstein n k x := by rw [Finset.mul_sum]
      -- In this step we increase the sum over `S` back to a sum over all of `Fin (n+1)`,
      -- so that we can use `bernstein.probability`.
      _ ≤ ε / 2 * ∑ k : Fin (n + 1), bernstein n k x := by
        gcongr
        exact Finset.sum_le_univ_sum_of_nonneg fun k => bernstein_nonneg
      _ = ε / 2 := by rw [bernstein.probability, mul_one]
  · -- We now turn to working on `Sᶜ`: we control the difference term just using `‖f‖`,
    -- and then insert a `δ^(-2) * (x - k/n)^2` factor
    -- (which is at least one because we are not in `S`).
    calc
      ∑ k in Sᶜ, |f k/ₙ - f x| * bernstein n k x ≤ ∑ k in Sᶜ, 2 * ‖f‖ * bernstein n k x := by
        gcongr
        apply f.dist_le_two_norm
      _ = 2 * ‖f‖ * ∑ k in Sᶜ, bernstein n k x := by rw [Finset.mul_sum]
      _ ≤ 2 * ‖f‖ * ∑ k in Sᶜ, δ ^ (-2 : ℤ) * ((x : ℝ) - k/ₙ) ^ 2 * bernstein n k x := by
        gcongr with _ m
        conv_lhs => rw [← one_mul (bernstein _ _ _)]
        gcongr
        exact le_of_mem_S_compl m
      -- Again enlarging the sum from `Sᶜ` to all of `Fin (n+1)`
      _ ≤ 2 * ‖f‖ * ∑ k : Fin (n + 1), δ ^ (-2 : ℤ) * ((x : ℝ) - k/ₙ) ^ 2 * bernstein n k x := by
        gcongr
        refine Finset.sum_le_univ_sum_of_nonneg <| fun k => ?_
        positivity
      _ = 2 * ‖f‖ * δ ^ (-2 : ℤ) * ∑ k : Fin (n + 1), ((x : ℝ) - k/ₙ) ^ 2 * bernstein n k x := by
        conv_rhs =>
          rw [mul_assoc, Finset.mul_sum]
          simp only [← mul_assoc]
      -- `bernstein.variance` and `x ∈ [0,1]` gives the uniform bound
      _ = 2 * ‖f‖ * δ ^ (-2 : ℤ) * x * (1 - x) / n := by rw [variance npos]; ring
      _ ≤ 2 * ‖f‖ * δ ^ (-2 : ℤ) * 1 * 1 / n := by 

Proof state:
f : C(↑I, ℝ)
ε : ℝ
h : 0 < ε
δ : ℝ := bernsteinApproximation.δ f ε h
nhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)
n : ℕ
nh : 2 * ‖f‖ * δ ^ (-2) / ↑n < ε / 2
npos' : 0 < n
npos : 0 < ↑n
x : ↑I
S : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x
⊢ 2 * ‖f‖ * δ ^ (-2) * ↑x * (1 - ↑x) / ↑n ≤ 2 * ‖f‖ * δ ^ (-2) * 1 * 1 / ↑n

Next tactic:
gcongr