/-- In finite dimensions, over an algebraically closed field, the generalized eigenspaces of any
linear endomorphism span the whole space. -/
theorem iSup_generalizedEigenspace_eq_top [IsAlgClosed K] [FiniteDimensional K V] (f : End K V) :
    ⨆ (μ : K) (k : ℕ), f.generalizedEigenspace μ k = ⊤ := by
  -- We prove the claim by strong induction on the dimension of the vector space.
  induction' h_dim : finrank K V using Nat.strong_induction_on with n ih generalizing V
  cases' n with n
  -- If the vector space is 0-dimensional, the result is trivial.
  · rw [← top_le_iff]
    simp only [Submodule.finrank_eq_zero.1 (Eq.trans (finrank_top _ _) h_dim), bot_le]
  -- Otherwise the vector space is nontrivial.
  · haveI : Nontrivial V := finrank_pos_iff.1 (by rw [h_dim]; apply Nat.zero_lt_succ)
    -- Hence, `f` has an eigenvalue `μ₀`.
    obtain ⟨μ₀, hμ₀⟩ : ∃ μ₀, f.HasEigenvalue μ₀ := exists_eigenvalue f
    -- We define `ES` to be the generalized eigenspace
    let ES := f.generalizedEigenspace μ₀ (finrank K V)
    -- and `ER` to be the generalized eigenrange.
    let ER := f.generalizedEigenrange μ₀ (finrank K V)
    -- `f` maps `ER` into itself.
    have h_f_ER : ∀ x : V, x ∈ ER → f x ∈ ER := fun x hx =>
      map_generalizedEigenrange_le (Submodule.mem_map_of_mem hx)
    -- Therefore, we can define the restriction `f'` of `f` to `ER`.
    let f' : End K ER := f.restrict h_f_ER
    -- The dimension of `ES` is positive
    have h_dim_ES_pos : 0 < finrank K ES := by
      dsimp only
      rw [h_dim]
      apply pos_finrank_generalizedEigenspace_of_hasEigenvalue hμ₀ (Nat.zero_lt_succ n)
    -- and the dimensions of `ES` and `ER` add up to `finrank K V`.
    have h_dim_add : finrank K ER + finrank K ES = finrank K V := by
      apply LinearMap.finrank_range_add_finrank_ker
    -- Therefore the dimension `ER` mus be smaller than `finrank K V`.
    have h_dim_ER : finrank K ER < n.succ := by linarith
    -- This allows us to apply the induction hypothesis on `ER`:
    have ih_ER : ⨆ (μ : K) (k : ℕ), f'.generalizedEigenspace μ k = ⊤ :=
      ih (finrank K ER) h_dim_ER f' rfl
    -- The induction hypothesis gives us a statement about subspaces of `ER`. We can transfer this
    -- to a statement about subspaces of `V` via `submodule.subtype`:
    have ih_ER' : ⨆ (μ : K) (k : ℕ), (f'.generalizedEigenspace μ k).map ER.subtype = ER := by
      simp only [(Submodule.map_iSup _ _).symm, ih_ER, Submodule.map_subtype_top ER]
    -- Moreover, every generalized eigenspace of `f'` is contained in the corresponding generalized
    -- eigenspace of `f`.
    have hff' :
      ∀ μ k, (f'.generalizedEigenspace μ k).map ER.subtype ≤ f.generalizedEigenspace μ k := by
      intros
      rw [generalizedEigenspace_restrict]
      apply Submodule.map_comap_le
    -- It follows that `ER` is contained in the span of all generalized eigenvectors.
    have hER : ER ≤ ⨆ (μ : K) (k : ℕ), f.generalizedEigenspace μ k := by
      rw [← ih_ER']
      exact iSup₂_mono hff'
    -- `ES` is contained in this span by definition.
    have hES : ES ≤ ⨆ (μ : K) (k : ℕ), f.generalizedEigenspace μ k :=
      le_trans (le_iSup (fun k => f.generalizedEigenspace μ₀ k) (finrank K V))
        (le_iSup (fun μ : K => ⨆ k : ℕ, f.generalizedEigenspace μ k) μ₀)
    -- Moreover, we know that `ER` and `ES` are disjoint.
    have h_disjoint : Disjoint ER ES := generalized_eigenvec_disjoint_range_ker f μ₀
    -- Since the dimensions of `ER` and `ES` add up to the dimension of `V`, it follows that the
    -- span of all generalized eigenvectors is all of `V`.
    show ⨆ (μ : K) (k : ℕ), f.generalizedEigenspace μ k = ⊤
    · rw [← top_le_iff, ← Submodule.eq_top_of_disjoint ER ES h_dim_add h_disjoint]
      

Proof state:
case h.succ.intro
K : Type u_1
inst✝⁴ : Field K
inst✝³ : IsAlgClosed K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
n : ℕ
ih :
  ∀ m < Nat.succ n,
    ∀ {V : Type u_2} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),
      finrank K V = m → ⨆ μ, ⨆ k, (generalizedEigenspace f μ) k = ⊤
h_dim : finrank K V = Nat.succ n
this : Nontrivial V
μ₀ : K
hμ₀ : HasEigenvalue f μ₀
ES : Submodule K V := (generalizedEigenspace f μ₀) (finrank K V)
ER : Submodule K V := generalizedEigenrange f μ₀ (finrank K V)
h_f_ER : ∀ x ∈ ER, f x ∈ ER
f' : End K ↥ER := LinearMap.restrict f h_f_ER
h_dim_ES_pos : 0 < finrank K ↥ES
h_dim_add : finrank K ↥ER + finrank K ↥ES = finrank K V
h_dim_ER : finrank K ↥ER < Nat.succ n
ih_ER : ⨆ μ, ⨆ k, (generalizedEigenspace f' μ) k = ⊤
ih_ER' : ⨆ μ, ⨆ k, Submodule.map (Submodule.subtype ER) ((generalizedEigenspace f' μ) k) = ER
hff' :
  ∀ (μ : K) (k : ℕ),
    Submodule.map (Submodule.subtype ER) ((generalizedEigenspace f' μ) k) ≤ (generalizedEigenspace f μ) k
hER : ER ≤ ⨆ μ, ⨆ k, (generalizedEigenspace f μ) k
hES : ES ≤ ⨆ μ, ⨆ k, (generalizedEigenspace f μ) k
h_disjoint : Disjoint ER ES
⊢ ER ⊔ ES ≤ ⨆ μ, ⨆ k, (generalizedEigenspace f μ) k

Next tactic:
apply sup_le hER hES