[GOAL]
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
⊢ NormedAddGroupHom.SurjectiveOnWith f (AddSubgroup.topologicalClosure K) (C + ε)
[PROOFSTEP]
rintro (h : H)
  (h_in : h ∈ K.topologicalClosure)
    -- We first get rid of the easy case where `h = 0`.
[GOAL]
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
⊢ ∃ g, ↑f g = h ∧ ‖g‖ ≤ (C + ε) * ‖h‖
[PROOFSTEP]
by_cases hyp_h : h = 0
[GOAL]
case pos
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : h = 0
⊢ ∃ g, ↑f g = h ∧ ‖g‖ ≤ (C + ε) * ‖h‖
[PROOFSTEP]
rw [hyp_h]
[GOAL]
case pos
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : h = 0
⊢ ∃ g, ↑f g = 0 ∧ ‖g‖ ≤ (C + ε) * ‖0‖
[PROOFSTEP]
use 0
[GOAL]
case h
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : h = 0
⊢ ↑f 0 = 0 ∧ ‖0‖ ≤ (C + ε) * ‖0‖
[PROOFSTEP]
simp
  /- The desired preimage will be constructed as the sum of a series. Convergence of
      the series will be guaranteed by completeness of `G`. We first write `h` as the sum
      of a sequence `v` of elements of `K` which starts close to `h` and then quickly goes to zero.
      The sequence `b` below quantifies this. -/
[GOAL]
case neg
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
⊢ ∃ g, ↑f g = h ∧ ‖g‖ ≤ (C + ε) * ‖h‖
[PROOFSTEP]
set b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
[GOAL]
case neg
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
⊢ ∃ g, ↑f g = h ∧ ‖g‖ ≤ (C + ε) * ‖h‖
[PROOFSTEP]
have b_pos : ∀ i, 0 < b i := by
  intro i
  field_simp [hC]
  exact div_pos (mul_pos hε (norm_pos_iff.mpr hyp_h)) (mul_pos (by norm_num : (0 : ℝ) < 2 ^ i * 2) hC)
[GOAL]
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
⊢ ∀ (i : ℕ), 0 < b i
[PROOFSTEP]
intro i
[GOAL]
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
i : ℕ
⊢ 0 < b i
[PROOFSTEP]
field_simp [hC]
[GOAL]
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
i : ℕ
⊢ 0 < ε * ‖h‖ / (2 ^ i * 2 * C)
[PROOFSTEP]
exact div_pos (mul_pos hε (norm_pos_iff.mpr hyp_h)) (mul_pos (by norm_num : (0 : ℝ) < 2 ^ i * 2) hC)
[GOAL]
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
i : ℕ
⊢ 0 < 2 ^ i * 2
[PROOFSTEP]
norm_num
[GOAL]
case neg
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
⊢ ∃ g, ↑f g = h ∧ ‖g‖ ≤ (C + ε) * ‖h‖
[PROOFSTEP]
obtain
  ⟨v : ℕ → H, lim_v : Tendsto (fun n : ℕ => ∑ k in range (n + 1), v k) atTop (𝓝 h), v_in : ∀ n, v n ∈ K, hv₀ :
    ‖v 0 - h‖ < b 0, hv : ∀ n > 0, ‖v n‖ < b n⟩ :=
  controlled_sum_of_mem_closure h_in b_pos
[GOAL]
case neg.intro.intro.intro.intro
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
⊢ ∃ g, ↑f g = h ∧ ‖g‖ ≤ (C + ε) * ‖h‖
[PROOFSTEP]
have : ∀ n, ∃ m' : G, f m' = v n ∧ ‖m'‖ ≤ C * ‖v n‖ := fun n : ℕ => hyp (v n) (v_in n)
[GOAL]
case neg.intro.intro.intro.intro
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
this : ∀ (n : ℕ), ∃ m', ↑f m' = v n ∧ ‖m'‖ ≤ C * ‖v n‖
⊢ ∃ g, ↑f g = h ∧ ‖g‖ ≤ (C + ε) * ‖h‖
[PROOFSTEP]
choose u hu hnorm_u using this
[GOAL]
case neg.intro.intro.intro.intro
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
⊢ ∃ g, ↑f g = h ∧ ‖g‖ ≤ (C + ε) * ‖h‖
[PROOFSTEP]
set s : ℕ → G := fun n => ∑ k in range (n + 1), u k
[GOAL]
case neg.intro.intro.intro.intro
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
⊢ ∃ g, ↑f g = h ∧ ‖g‖ ≤ (C + ε) * ‖h‖
[PROOFSTEP]
have : CauchySeq s :=
  by
  apply NormedAddCommGroup.cauchy_series_of_le_geometric'' (by norm_num) one_half_lt_one
  rintro n (hn : n ≥ 1)
  calc
    ‖u n‖ ≤ C * ‖v n‖ := hnorm_u n
    _ ≤ C * b n := by gcongr; exact (hv _ <| Nat.succ_le_iff.mp hn).le
    _ = (1 / 2) ^ n * (ε * ‖h‖ / 2) := by simp [mul_div_cancel' _ hC.ne.symm]
    _ = ε * ‖h‖ / 2 * (1 / 2) ^ n :=
      mul_comm _
        _
          -- We now show that the limit `g` of `s` is the desired preimage.
[GOAL]
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
⊢ CauchySeq s
[PROOFSTEP]
apply NormedAddCommGroup.cauchy_series_of_le_geometric'' (by norm_num) one_half_lt_one
[GOAL]
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
⊢ 0 < 1 / 2
[PROOFSTEP]
norm_num
[GOAL]
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
⊢ ∀ (n : ℕ), n ≥ ?m.14929 → ‖u n‖ ≤ ?m.14927 * (1 / 2) ^ n
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
⊢ ℝ
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
⊢ ℕ
[PROOFSTEP]
rintro n (hn : n ≥ 1)
[GOAL]
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
n : ℕ
hn : n ≥ 1
⊢ ‖u n‖ ≤ ?m.14927 * (1 / 2) ^ n
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
⊢ ℝ
[PROOFSTEP]
calc
  ‖u n‖ ≤ C * ‖v n‖ := hnorm_u n
  _ ≤ C * b n := by gcongr; exact (hv _ <| Nat.succ_le_iff.mp hn).le
  _ = (1 / 2) ^ n * (ε * ‖h‖ / 2) := by simp [mul_div_cancel' _ hC.ne.symm]
  _ = ε * ‖h‖ / 2 * (1 / 2) ^ n :=
    mul_comm _
      _
        -- We now show that the limit `g` of `s` is the desired preimage.
[GOAL]
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
n : ℕ
hn : n ≥ 1
⊢ C * ‖v n‖ ≤ C * b n
[PROOFSTEP]
gcongr
[GOAL]
case h
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
n : ℕ
hn : n ≥ 1
⊢ ‖v n‖ ≤ b n
[PROOFSTEP]
exact (hv _ <| Nat.succ_le_iff.mp hn).le
[GOAL]
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
n : ℕ
hn : n ≥ 1
⊢ C * b n = (1 / 2) ^ n * (ε * ‖h‖ / 2)
[PROOFSTEP]
simp [mul_div_cancel' _ hC.ne.symm]
[GOAL]
case neg.intro.intro.intro.intro
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this : CauchySeq s
⊢ ∃ g, ↑f g = h ∧ ‖g‖ ≤ (C + ε) * ‖h‖
[PROOFSTEP]
obtain ⟨g : G, hg⟩ := cauchySeq_tendsto_of_complete this
[GOAL]
case neg.intro.intro.intro.intro.intro
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
⊢ ∃ g, ↑f g = h ∧ ‖g‖ ≤ (C + ε) * ‖h‖
[PROOFSTEP]
refine' ⟨g, _, _⟩
[GOAL]
case neg.intro.intro.intro.intro.intro.refine'_1
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
⊢ ↑f g = h
[PROOFSTEP]
have : f ∘ s = fun n => ∑ k in range (n + 1), v k := by
  ext n
  simp [map_sum, hu]
    /- In the above equality, the left-hand-side converges to `f g` by continuity of `f` and
          definition of `g` while the right-hand-side converges to `h` by construction of `v` so
          `g` is indeed a preimage of `h`. -/
[GOAL]
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
⊢ ↑f ∘ s = fun n => ∑ k in range (n + 1), v k
[PROOFSTEP]
ext n
[GOAL]
case h
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
n : ℕ
⊢ (↑f ∘ s) n = ∑ k in range (n + 1), v k
[PROOFSTEP]
simp [map_sum, hu]
  /- In the above equality, the left-hand-side converges to `f g` by continuity of `f` and
        definition of `g` while the right-hand-side converges to `h` by construction of `v` so
        `g` is indeed a preimage of `h`. -/
[GOAL]
case neg.intro.intro.intro.intro.intro.refine'_1
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this✝ : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
this : ↑f ∘ s = fun n => ∑ k in range (n + 1), v k
⊢ ↑f g = h
[PROOFSTEP]
rw [← this] at lim_v 
[GOAL]
case neg.intro.intro.intro.intro.intro.refine'_1
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
lim_v : Tendsto (↑f ∘ s) atTop (𝓝 h)
this✝ : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
this : ↑f ∘ s = fun n => ∑ k in range (n + 1), v k
⊢ ↑f g = h
[PROOFSTEP]
exact tendsto_nhds_unique ((f.continuous.tendsto g).comp hg) lim_v
[GOAL]
case neg.intro.intro.intro.intro.intro.refine'_2
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
⊢ ‖g‖ ≤ (C + ε) * ‖h‖
[PROOFSTEP]
suffices : ∀ n, ‖s n‖ ≤ (C + ε) * ‖h‖
[GOAL]
case neg.intro.intro.intro.intro.intro.refine'_2
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this✝ : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
this : ∀ (n : ℕ), ‖s n‖ ≤ (C + ε) * ‖h‖
⊢ ‖g‖ ≤ (C + ε) * ‖h‖
case this
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
⊢ ∀ (n : ℕ), ‖s n‖ ≤ (C + ε) * ‖h‖
[PROOFSTEP]
exact le_of_tendsto' (continuous_norm.continuousAt.tendsto.comp hg) this
[GOAL]
case this
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
⊢ ∀ (n : ℕ), ‖s n‖ ≤ (C + ε) * ‖h‖
[PROOFSTEP]
intro n
[GOAL]
case this
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
n : ℕ
⊢ ‖s n‖ ≤ (C + ε) * ‖h‖
[PROOFSTEP]
have hnorm₀ : ‖u 0‖ ≤ C * b 0 + C * ‖h‖ :=
  by
  have :=
    calc
      ‖v 0‖ ≤ ‖h‖ + ‖v 0 - h‖ := norm_le_insert' _ _
      _ ≤ ‖h‖ + b 0 := by gcongr
  calc
    ‖u 0‖ ≤ C * ‖v 0‖ := hnorm_u 0
    _ ≤ C * (‖h‖ + b 0) := by gcongr
    _ = C * b 0 + C * ‖h‖ := by rw [add_comm, mul_add]
[GOAL]
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
n : ℕ
⊢ ‖u 0‖ ≤ C * b 0 + C * ‖h‖
[PROOFSTEP]
have :=
  calc
    ‖v 0‖ ≤ ‖h‖ + ‖v 0 - h‖ := norm_le_insert' _ _
    _ ≤ ‖h‖ + b 0 := by gcongr
[GOAL]
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
n : ℕ
⊢ ‖h‖ + ‖v 0 - h‖ ≤ ‖h‖ + b 0
[PROOFSTEP]
gcongr
[GOAL]
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this✝ : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
n : ℕ
this : ‖v 0‖ ≤ ‖h‖ + b 0
⊢ ‖u 0‖ ≤ C * b 0 + C * ‖h‖
[PROOFSTEP]
calc
  ‖u 0‖ ≤ C * ‖v 0‖ := hnorm_u 0
  _ ≤ C * (‖h‖ + b 0) := by gcongr
  _ = C * b 0 + C * ‖h‖ := by rw [add_comm, mul_add]
[GOAL]
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this✝ : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
n : ℕ
this : ‖v 0‖ ≤ ‖h‖ + b 0
⊢ C * ‖v 0‖ ≤ C * (‖h‖ + b 0)
[PROOFSTEP]
gcongr
[GOAL]
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this✝ : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
n : ℕ
this : ‖v 0‖ ≤ ‖h‖ + b 0
⊢ C * (‖h‖ + b 0) = C * b 0 + C * ‖h‖
[PROOFSTEP]
rw [add_comm, mul_add]
[GOAL]
case this
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
n : ℕ
hnorm₀ : ‖u 0‖ ≤ C * b 0 + C * ‖h‖
⊢ ‖s n‖ ≤ (C + ε) * ‖h‖
[PROOFSTEP]
have : (∑ k in range (n + 1), C * b k) ≤ ε * ‖h‖ :=
  calc
    (∑ k in range (n + 1), C * b k) = (∑ k in range (n + 1), (1 / 2 : ℝ) ^ k) * (ε * ‖h‖ / 2) := by
      simp only [mul_div_cancel' _ hC.ne.symm, ← sum_mul]
    _ ≤ 2 * (ε * ‖h‖ / 2) := by gcongr; apply sum_geometric_two_le
    _ = ε * ‖h‖ := mul_div_cancel' _ two_ne_zero
[GOAL]
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
n : ℕ
hnorm₀ : ‖u 0‖ ≤ C * b 0 + C * ‖h‖
⊢ ∑ k in range (n + 1), C * b k = (∑ k in range (n + 1), (1 / 2) ^ k) * (ε * ‖h‖ / 2)
[PROOFSTEP]
simp only [mul_div_cancel' _ hC.ne.symm, ← sum_mul]
[GOAL]
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
n : ℕ
hnorm₀ : ‖u 0‖ ≤ C * b 0 + C * ‖h‖
⊢ (∑ k in range (n + 1), (1 / 2) ^ k) * (ε * ‖h‖ / 2) ≤ 2 * (ε * ‖h‖ / 2)
[PROOFSTEP]
gcongr
[GOAL]
case h
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
n : ℕ
hnorm₀ : ‖u 0‖ ≤ C * b 0 + C * ‖h‖
⊢ ∑ k in range (n + 1), (1 / 2) ^ k ≤ 2
[PROOFSTEP]
apply sum_geometric_two_le
[GOAL]
case this
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this✝ : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
n : ℕ
hnorm₀ : ‖u 0‖ ≤ C * b 0 + C * ‖h‖
this : ∑ k in range (n + 1), C * b k ≤ ε * ‖h‖
⊢ ‖s n‖ ≤ (C + ε) * ‖h‖
[PROOFSTEP]
calc
  ‖s n‖ ≤ ∑ k in range (n + 1), ‖u k‖ := norm_sum_le _ _
  _ = (∑ k in range n, ‖u (k + 1)‖) + ‖u 0‖ := (sum_range_succ' _ _)
  _ ≤ (∑ k in range n, C * ‖v (k + 1)‖) + ‖u 0‖ := by gcongr; apply hnorm_u
  _ ≤ (∑ k in range n, C * b (k + 1)) + (C * b 0 + C * ‖h‖) := by gcongr with k; exact (hv _ k.succ_pos).le
  _ = (∑ k in range (n + 1), C * b k) + C * ‖h‖ := by rw [← add_assoc, sum_range_succ']
  _ ≤ (C + ε) * ‖h‖ := by
    rw [add_comm, add_mul]
    apply add_le_add_left this
[GOAL]
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this✝ : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
n : ℕ
hnorm₀ : ‖u 0‖ ≤ C * b 0 + C * ‖h‖
this : ∑ k in range (n + 1), C * b k ≤ ε * ‖h‖
⊢ ∑ k in range n, ‖u (k + 1)‖ + ‖u 0‖ ≤ ∑ k in range n, C * ‖v (k + 1)‖ + ‖u 0‖
[PROOFSTEP]
gcongr
[GOAL]
case bc.h
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this✝ : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
n : ℕ
hnorm₀ : ‖u 0‖ ≤ C * b 0 + C * ‖h‖
this : ∑ k in range (n + 1), C * b k ≤ ε * ‖h‖
i✝ : ℕ
a✝ : i✝ ∈ range n
⊢ ‖u (i✝ + 1)‖ ≤ C * ‖v (i✝ + 1)‖
[PROOFSTEP]
apply hnorm_u
[GOAL]
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this✝ : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
n : ℕ
hnorm₀ : ‖u 0‖ ≤ C * b 0 + C * ‖h‖
this : ∑ k in range (n + 1), C * b k ≤ ε * ‖h‖
⊢ ∑ k in range n, C * ‖v (k + 1)‖ + ‖u 0‖ ≤ ∑ k in range n, C * b (k + 1) + (C * b 0 + C * ‖h‖)
[PROOFSTEP]
gcongr with k
[GOAL]
case h₁.h.h
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this✝ : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
n : ℕ
hnorm₀ : ‖u 0‖ ≤ C * b 0 + C * ‖h‖
this : ∑ k in range (n + 1), C * b k ≤ ε * ‖h‖
k : ℕ
a✝ : k ∈ range n
⊢ ‖v (k + 1)‖ ≤ b (k + 1)
[PROOFSTEP]
exact (hv _ k.succ_pos).le
[GOAL]
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this✝ : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
n : ℕ
hnorm₀ : ‖u 0‖ ≤ C * b 0 + C * ‖h‖
this : ∑ k in range (n + 1), C * b k ≤ ε * ‖h‖
⊢ ∑ k in range n, C * b (k + 1) + (C * b 0 + C * ‖h‖) = ∑ k in range (n + 1), C * b k + C * ‖h‖
[PROOFSTEP]
rw [← add_assoc, sum_range_succ']
[GOAL]
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this✝ : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
n : ℕ
hnorm₀ : ‖u 0‖ ≤ C * b 0 + C * ‖h‖
this : ∑ k in range (n + 1), C * b k ≤ ε * ‖h‖
⊢ ∑ k in range (n + 1), C * b k + C * ‖h‖ ≤ (C + ε) * ‖h‖
[PROOFSTEP]
rw [add_comm, add_mul]
[GOAL]
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : NormedAddGroupHom.SurjectiveOnWith f K C
h : H
h_in : h ∈ AddSubgroup.topologicalClosure K
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i => (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n => ∑ k in range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖v 0 - h‖ < b 0
hv : ∀ (n : ℕ), n > 0 → ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), ↑f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n => ∑ k in range (n + 1), u k
this✝ : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
n : ℕ
hnorm₀ : ‖u 0‖ ≤ C * b 0 + C * ‖h‖
this : ∑ k in range (n + 1), C * b k ≤ ε * ‖h‖
⊢ C * ‖h‖ + ∑ k in range (n + 1), C * b k ≤ C * ‖h‖ + ε * ‖h‖
[PROOFSTEP]
apply add_le_add_left this
[GOAL]
G : Type u_1
inst✝³ : NormedAddCommGroup G
inst✝² : CompleteSpace G
H : Type u_2
inst✝¹ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : Type u_3
inst✝ : SeminormedAddCommGroup K
j : NormedAddGroupHom K H
hj : ∀ (x : K), ‖↑j x‖ = ‖x‖
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : ∀ (k : K), ∃ g, ↑f g = ↑j k ∧ ‖g‖ ≤ C * ‖k‖
⊢ NormedAddGroupHom.SurjectiveOnWith f (AddSubgroup.topologicalClosure (NormedAddGroupHom.range j)) (C + ε)
[PROOFSTEP]
replace hyp : ∀ h ∈ j.range, ∃ g, f g = h ∧ ‖g‖ ≤ C * ‖h‖
[GOAL]
case hyp
G : Type u_1
inst✝³ : NormedAddCommGroup G
inst✝² : CompleteSpace G
H : Type u_2
inst✝¹ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : Type u_3
inst✝ : SeminormedAddCommGroup K
j : NormedAddGroupHom K H
hj : ∀ (x : K), ‖↑j x‖ = ‖x‖
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : ∀ (k : K), ∃ g, ↑f g = ↑j k ∧ ‖g‖ ≤ C * ‖k‖
⊢ ∀ (h : H), h ∈ NormedAddGroupHom.range j → ∃ g, ↑f g = h ∧ ‖g‖ ≤ C * ‖h‖
[PROOFSTEP]
intro h h_in
[GOAL]
case hyp
G : Type u_1
inst✝³ : NormedAddCommGroup G
inst✝² : CompleteSpace G
H : Type u_2
inst✝¹ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : Type u_3
inst✝ : SeminormedAddCommGroup K
j : NormedAddGroupHom K H
hj : ∀ (x : K), ‖↑j x‖ = ‖x‖
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : ∀ (k : K), ∃ g, ↑f g = ↑j k ∧ ‖g‖ ≤ C * ‖k‖
h : H
h_in : h ∈ NormedAddGroupHom.range j
⊢ ∃ g, ↑f g = h ∧ ‖g‖ ≤ C * ‖h‖
[PROOFSTEP]
rcases(j.mem_range _).mp h_in with ⟨k, rfl⟩
[GOAL]
case hyp.intro
G : Type u_1
inst✝³ : NormedAddCommGroup G
inst✝² : CompleteSpace G
H : Type u_2
inst✝¹ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : Type u_3
inst✝ : SeminormedAddCommGroup K
j : NormedAddGroupHom K H
hj : ∀ (x : K), ‖↑j x‖ = ‖x‖
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : ∀ (k : K), ∃ g, ↑f g = ↑j k ∧ ‖g‖ ≤ C * ‖k‖
k : K
h_in : ↑j k ∈ NormedAddGroupHom.range j
⊢ ∃ g, ↑f g = ↑j k ∧ ‖g‖ ≤ C * ‖↑j k‖
[PROOFSTEP]
rw [hj]
[GOAL]
case hyp.intro
G : Type u_1
inst✝³ : NormedAddCommGroup G
inst✝² : CompleteSpace G
H : Type u_2
inst✝¹ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : Type u_3
inst✝ : SeminormedAddCommGroup K
j : NormedAddGroupHom K H
hj : ∀ (x : K), ‖↑j x‖ = ‖x‖
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : ∀ (k : K), ∃ g, ↑f g = ↑j k ∧ ‖g‖ ≤ C * ‖k‖
k : K
h_in : ↑j k ∈ NormedAddGroupHom.range j
⊢ ∃ g, ↑f g = ↑j k ∧ ‖g‖ ≤ C * ‖k‖
[PROOFSTEP]
exact hyp k
[GOAL]
G : Type u_1
inst✝³ : NormedAddCommGroup G
inst✝² : CompleteSpace G
H : Type u_2
inst✝¹ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : Type u_3
inst✝ : SeminormedAddCommGroup K
j : NormedAddGroupHom K H
hj : ∀ (x : K), ‖↑j x‖ = ‖x‖
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : ∀ (h : H), h ∈ NormedAddGroupHom.range j → ∃ g, ↑f g = h ∧ ‖g‖ ≤ C * ‖h‖
⊢ NormedAddGroupHom.SurjectiveOnWith f (AddSubgroup.topologicalClosure (NormedAddGroupHom.range j)) (C + ε)
[PROOFSTEP]
exact controlled_closure_of_complete hC hε hyp
