/-- If a sequence of functions real or complex functions are eventually differentiable on a
neighborhood of `x`, they are Cauchy _at_ `x`, and their derivatives
are a uniform Cauchy sequence in a neighborhood of `x`, then the functions form a uniform Cauchy
sequence in a neighborhood of `x`. -/
theorem uniformCauchySeqOnFilter_of_fderiv (hf' : UniformCauchySeqOnFilter f' l (𝓝 x))
    (hf : ∀ᶠ n : ι × E in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2)
    (hfg : Cauchy (map (fun n => f n x) l)) : UniformCauchySeqOnFilter f l (𝓝 x) := by
  letI : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 _
  rw [SeminormedAddGroup.uniformCauchySeqOnFilter_iff_tendstoUniformlyOnFilter_zero] at hf' ⊢
  suffices
    TendstoUniformlyOnFilter (fun (n : ι × ι) (z : E) => f n.1 z - f n.2 z - (f n.1 x - f n.2 x)) 0
        (l ×ˢ l) (𝓝 x) ∧
      TendstoUniformlyOnFilter (fun (n : ι × ι) (_ : E) => f n.1 x - f n.2 x) 0 (l ×ˢ l) (𝓝 x) by
    have := this.1.add this.2
    rw [add_zero] at this
    exact this.congr (by simp)
  constructor
  · -- This inequality follows from the mean value theorem. To apply it, we will need to shrink our
    -- neighborhood to small enough ball
    rw [Metric.tendstoUniformlyOnFilter_iff] at hf' ⊢
    intro ε hε
    have := (tendsto_swap4_prod.eventually (hf.prod_mk hf)).diag_of_prod_right
    obtain ⟨a, b, c, d, e⟩ := eventually_prod_iff.1 ((hf' ε hε).and this)
    obtain ⟨R, hR, hR'⟩ := Metric.nhds_basis_ball.eventually_iff.mp d
    let r := min 1 R
    have hr : 0 < r := by simp [hR]
    have hr' : ∀ ⦃y : E⦄, y ∈ Metric.ball x r → c y := fun y hy =>
      hR' (lt_of_lt_of_le (Metric.mem_ball.mp hy) (min_le_right _ _))
    have hxy : ∀ y : E, y ∈ Metric.ball x r → ‖y - x‖ < 1 := by
      intro y hy
      rw [Metric.mem_ball, dist_eq_norm] at hy
      exact lt_of_lt_of_le hy (min_le_left _ _)
    have hxyε : ∀ y : E, y ∈ Metric.ball x r → ε * ‖y - x‖ < ε := by
      intro y hy
      exact (mul_lt_iff_lt_one_right hε.lt).mpr (hxy y hy)
    -- With a small ball in hand, apply the mean value theorem
    refine'
      eventually_prod_iff.mpr
        ⟨_, b, fun e : E => Metric.ball x r e,
          eventually_mem_set.mpr (Metric.nhds_basis_ball.mem_of_mem hr), fun {n} hn {y} hy => _⟩
    simp only [Pi.zero_apply, dist_zero_left] at e ⊢
    

Proof state:
case left.intro.intro.intro.intro.intro.intro
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
x : E
hf' : ∀ ε > 0, ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (OfNat.ofNat 0 n.2) (f' n.1.1 n.2 - f' n.1.2 n.2) < ε
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : Cauchy (map (fun n => f n x) l)
this✝ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
ε : ℝ
hε : ε > 0
this :
  ∀ᶠ (x : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x,
    HasFDerivAt (f (((x.1, x.2, x.2).1.1, (x.1, x.2, x.2).2.1), (x.1, x.2, x.2).1.2, (x.1, x.2, x.2).2.2).1.1)
        (f' (((x.1, x.2, x.2).1.1, (x.1, x.2, x.2).2.1), (x.1, x.2, x.2).1.2, (x.1, x.2, x.2).2.2).1.1
          (((x.1, x.2, x.2).1.1, (x.1, x.2, x.2).2.1), (x.1, x.2, x.2).1.2, (x.1, x.2, x.2).2.2).1.2)
        (((x.1, x.2, x.2).1.1, (x.1, x.2, x.2).2.1), (x.1, x.2, x.2).1.2, (x.1, x.2, x.2).2.2).1.2 ∧
      HasFDerivAt (f (((x.1, x.2, x.2).1.1, (x.1, x.2, x.2).2.1), (x.1, x.2, x.2).1.2, (x.1, x.2, x.2).2.2).2.1)
        (f' (((x.1, x.2, x.2).1.1, (x.1, x.2, x.2).2.1), (x.1, x.2, x.2).1.2, (x.1, x.2, x.2).2.2).2.1
          (((x.1, x.2, x.2).1.1, (x.1, x.2, x.2).2.1), (x.1, x.2, x.2).1.2, (x.1, x.2, x.2).2.2).2.2)
        (((x.1, x.2, x.2).1.1, (x.1, x.2, x.2).2.1), (x.1, x.2, x.2).1.2, (x.1, x.2, x.2).2.2).2.2
a : ι × ι → Prop
b : ∀ᶠ (x : ι × ι) in l ×ˢ l, a x
c : E → Prop
d : ∀ᶠ (y : E) in 𝓝 x, c y
R : ℝ
hR : 0 < R
hR' : ∀ ⦃x_1 : E⦄, x_1 ∈ Metric.ball x R → c x_1
r : ℝ := min 1 R
hr : 0 < r
hr' : ∀ ⦃y : E⦄, y ∈ Metric.ball x r → c y
hxy : ∀ y ∈ Metric.ball x r, ‖y - x‖ < 1
hxyε : ∀ y ∈ Metric.ball x r, ε * ‖y - x‖ < ε
n : ι × ι
hn : a n
y : E
hy : (fun e => Metric.ball x r e) y
e :
  ∀ {x : ι × ι},
    a x →
      ∀ {y : E}, c y → ‖f' x.1 y - f' x.2 y‖ < ε ∧ HasFDerivAt (f x.1) (f' x.1 y) y ∧ HasFDerivAt (f x.2) (f' x.2 y) y
⊢ ‖f n.1 y - f n.2 y - (f n.1 x - f n.2 x)‖ < ε

Next tactic:
refine' lt_of_le_of_lt _ (hxyε y hy)