l3lab/ntp-mathlib huggingface dataset:
@[to_additive]
theorem controlled_prod_of_mem_closure {s : Subgroup E} (hg : a ∈ closure (s : Set E)) {b : ℕ → ℝ}
    (b_pos : ∀ n, 0 < b n) :
    ∃ v : ℕ → E,
      Tendsto (fun n => ∏ i in range (n + 1), v i) atTop (𝓝 a) ∧
        (∀ n, v n ∈ s) ∧ ‖v 0 / a‖ < b 0 ∧ ∀ n, 0 < n → ‖v n‖ < b n := by
  obtain ⟨u : ℕ → E, u_in : ∀ n, u n ∈ s, lim_u : Tendsto u atTop (𝓝 a)⟩ :=
    mem_closure_iff_seq_limit.mp hg
  obtain ⟨n₀, hn₀⟩ : ∃ n₀, ∀ n ≥ n₀, ‖u n / a‖ < b 0 :=
    haveI : { x | ‖x / a‖ < b 0 } ∈ 𝓝 a := by
      simp_rw [← dist_eq_norm_div]
      exact Metric.ball_mem_nhds _ (b_pos _)
    Filter.tendsto_atTop'.mp lim_u _ this
  set z : ℕ → E := fun n => u (n + n₀)
  have lim_z : Tendsto z atTop (𝓝 a) := lim_u.comp (tendsto_add_atTop_nat n₀)
  have mem_𝓤 : ∀ n, { p : E × E | ‖p.1 / p.2‖ < b (n + 1) } ∈ 𝓤 E := fun n => by
    simpa [← dist_eq_norm_div] using Metric.dist_mem_uniformity (b_pos <| n + 1)
  obtain ⟨φ : ℕ → ℕ, φ_extr : StrictMono φ, hφ : ∀ n, ‖z (φ <| n + 1) / z (φ n)‖ < b (n + 1)⟩ :=
    lim_z.cauchySeq.subseq_mem mem_𝓤
  set w : ℕ → E := z ∘ φ
  have hw : Tendsto w atTop (𝓝 a) := lim_z.comp φ_extr.tendsto_atTop
  set v : ℕ → E := fun i => if i = 0 then w 0 else w i / w (i - 1)
  refine' ⟨v, Tendsto.congr (Finset.eq_prod_range_div' w) hw, _, hn₀ _ (n₀.le_add_left _), _⟩
  · rintro ⟨⟩
    · change w 0 ∈ s
      

Proof state:
case intro.intro.intro.intro.intro.refine'_1.zero
𝓕 : Type u_1
𝕜 : Type u_2
α : Type u_3
ι : Type u_4
κ : Type u_5
E : Type u_6
F : Type u_7
G : Type u_8
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
a a₁ a₂ b✝ b₁ b₂ : E
r r₁ r₂ : ℝ
s : Subgroup E
hg : a ∈ closure ↑s
b : ℕ → ℝ
b_pos : ∀ (n : ℕ), 0 < b n
u : ℕ → E
u_in : ∀ (n : ℕ), u n ∈ s
lim_u : Tendsto u atTop (𝓝 a)
n₀ : ℕ
hn₀ : ∀ n ≥ n₀, ‖u n / a‖ < b 0
z : ℕ → E := fun n => u (n + n₀)
lim_z : Tendsto z atTop (𝓝 a)
mem_𝓤 : ∀ (n : ℕ), {p | ‖p.1 / p.2‖ < b (n + 1)} ∈ 𝓤 E
φ : ℕ → ℕ
φ_extr : StrictMono φ
hφ : ∀ (n : ℕ), ‖z (φ (n + 1)) / z (φ n)‖ < b (n + 1)
w : ℕ → E := z ∘ φ
hw : Tendsto w atTop (𝓝 a)
v : ℕ → E := fun i => if i = 0 then w 0 else w i / w (i - 1)
⊢ w 0 ∈ s

Next tactic:
apply u_in
@[to_additive]
theorem controlled_prod_of_mem_closure {s : Subgroup E} (hg : a ∈ closure (s : Set E)) {b : ℕ → ℝ}
    (b_pos : ∀ n, 0 < b n) :
    ∃ v : ℕ → E,
      Tendsto (fun n => ∏ i in range (n + 1), v i) atTop (𝓝 a) ∧
        (∀ n, v n ∈ s) ∧ ‖v 0 / a‖ < b 0 ∧ ∀ n, 0 < n → ‖v n‖ < b n := by
  obtain ⟨u : ℕ → E, u_in : ∀ n, u n ∈ s, lim_u : Tendsto u atTop (𝓝 a)⟩ :=
    mem_closure_iff_seq_limit.mp hg
  obtain ⟨n₀, hn₀⟩ : ∃ n₀, ∀ n ≥ n₀, ‖u n / a‖ < b 0 :=
    haveI : { x | ‖x / a‖ < b 0 } ∈ 𝓝 a := by
      simp_rw [← dist_eq_norm_div]
      exact Metric.ball_mem_nhds _ (b_pos _)
    Filter.tendsto_atTop'.mp lim_u _ this
  set z : ℕ → E := fun n => u (n + n₀)
  have lim_z : Tendsto z atTop (𝓝 a) := lim_u.comp (tendsto_add_atTop_nat n₀)
  have mem_𝓤 : ∀ n, { p : E × E | ‖p.1 / p.2‖ < b (n + 1) } ∈ 𝓤 E := fun n => by
    simpa [← dist_eq_norm_div] using Metric.dist_mem_uniformity (b_pos <| n + 1)
  obtain ⟨φ : ℕ → ℕ, φ_extr : StrictMono φ, hφ : ∀ n, ‖z (φ <| n + 1) / z (φ n)‖ < b (n + 1)⟩ :=
    lim_z.cauchySeq.subseq_mem mem_𝓤
  set w : ℕ → E := z ∘ φ
  have hw : Tendsto w atTop (𝓝 a) := lim_z.comp φ_extr.tendsto_atTop
  set v : ℕ → E := fun i => if i = 0 then w 0 else w i / w (i - 1)
  refine' ⟨v, Tendsto.congr (Finset.eq_prod_range_div' w) hw, _, hn₀ _ (n₀.le_add_left _), _⟩
  · rintro ⟨⟩
    · change w 0 ∈ s
      apply u_in
    · 

Proof state:
case intro.intro.intro.intro.intro.refine'_1.succ
𝓕 : Type u_1
𝕜 : Type u_2
α : Type u_3
ι : Type u_4
κ : Type u_5
E : Type u_6
F : Type u_7
G : Type u_8
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
a a₁ a₂ b✝ b₁ b₂ : E
r r₁ r₂ : ℝ
s : Subgroup E
hg : a ∈ closure ↑s
b : ℕ → ℝ
b_pos : ∀ (n : ℕ), 0 < b n
u : ℕ → E
u_in : ∀ (n : ℕ), u n ∈ s
lim_u : Tendsto u atTop (𝓝 a)
n₀ : ℕ
hn₀ : ∀ n ≥ n₀, ‖u n / a‖ < b 0
z : ℕ → E := fun n => u (n + n₀)
lim_z : Tendsto z atTop (𝓝 a)
mem_𝓤 : ∀ (n : ℕ), {p | ‖p.1 / p.2‖ < b (n + 1)} ∈ 𝓤 E
φ : ℕ → ℕ
φ_extr : StrictMono φ
hφ : ∀ (n : ℕ), ‖z (φ (n + 1)) / z (φ n)‖ < b (n + 1)
w : ℕ → E := z ∘ φ
hw : Tendsto w atTop (𝓝 a)
v : ℕ → E := fun i => if i = 0 then w 0 else w i / w (i - 1)
n✝ : ℕ
⊢ v (Nat.succ n✝) ∈ s

Next tactic:
apply s.div_mem
@[to_additive]
theorem controlled_prod_of_mem_closure {s : Subgroup E} (hg : a ∈ closure (s : Set E)) {b : ℕ → ℝ}
    (b_pos : ∀ n, 0 < b n) :
    ∃ v : ℕ → E,
      Tendsto (fun n => ∏ i in range (n + 1), v i) atTop (𝓝 a) ∧
        (∀ n, v n ∈ s) ∧ ‖v 0 / a‖ < b 0 ∧ ∀ n, 0 < n → ‖v n‖ < b n := by
  obtain ⟨u : ℕ → E, u_in : ∀ n, u n ∈ s, lim_u : Tendsto u atTop (𝓝 a)⟩ :=
    mem_closure_iff_seq_limit.mp hg
  obtain ⟨n₀, hn₀⟩ : ∃ n₀, ∀ n ≥ n₀, ‖u n / a‖ < b 0 :=
    haveI : { x | ‖x / a‖ < b 0 } ∈ 𝓝 a := by
      simp_rw [← dist_eq_norm_div]
      exact Metric.ball_mem_nhds _ (b_pos _)
    Filter.tendsto_atTop'.mp lim_u _ this
  set z : ℕ → E := fun n => u (n + n₀)
  have lim_z : Tendsto z atTop (𝓝 a) := lim_u.comp (tendsto_add_atTop_nat n₀)
  have mem_𝓤 : ∀ n, { p : E × E | ‖p.1 / p.2‖ < b (n + 1) } ∈ 𝓤 E := fun n => by
    simpa [← dist_eq_norm_div] using Metric.dist_mem_uniformity (b_pos <| n + 1)
  obtain ⟨φ : ℕ → ℕ, φ_extr : StrictMono φ, hφ : ∀ n, ‖z (φ <| n + 1) / z (φ n)‖ < b (n + 1)⟩ :=
    lim_z.cauchySeq.subseq_mem mem_𝓤
  set w : ℕ → E := z ∘ φ
  have hw : Tendsto w atTop (𝓝 a) := lim_z.comp φ_extr.tendsto_atTop
  set v : ℕ → E := fun i => if i = 0 then w 0 else w i / w (i - 1)
  refine' ⟨v, Tendsto.congr (Finset.eq_prod_range_div' w) hw, _, hn₀ _ (n₀.le_add_left _), _⟩
  · rintro ⟨⟩
    · change w 0 ∈ s
      apply u_in
    · apply s.div_mem <;> 

Proof state:
case intro.intro.intro.intro.intro.refine'_1.succ.hx
𝓕 : Type u_1
𝕜 : Type u_2
α : Type u_3
ι : Type u_4
κ : Type u_5
E : Type u_6
F : Type u_7
G : Type u_8
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
a a₁ a₂ b✝ b₁ b₂ : E
r r₁ r₂ : ℝ
s : Subgroup E
hg : a ∈ closure ↑s
b : ℕ → ℝ
b_pos : ∀ (n : ℕ), 0 < b n
u : ℕ → E
u_in : ∀ (n : ℕ), u n ∈ s
lim_u : Tendsto u atTop (𝓝 a)
n₀ : ℕ
hn₀ : ∀ n ≥ n₀, ‖u n / a‖ < b 0
z : ℕ → E := fun n => u (n + n₀)
lim_z : Tendsto z atTop (𝓝 a)
mem_𝓤 : ∀ (n : ℕ), {p | ‖p.1 / p.2‖ < b (n + 1)} ∈ 𝓤 E
φ : ℕ → ℕ
φ_extr : StrictMono φ
hφ : ∀ (n : ℕ), ‖z (φ (n + 1)) / z (φ n)‖ < b (n + 1)
w : ℕ → E := z ∘ φ
hw : Tendsto w atTop (𝓝 a)
v : ℕ → E := fun i => if i = 0 then w 0 else w i / w (i - 1)
n✝ : ℕ
⊢ w (Nat.succ n✝) ∈ s

Next tactic:
apply u_in
@[to_additive]
theorem controlled_prod_of_mem_closure {s : Subgroup E} (hg : a ∈ closure (s : Set E)) {b : ℕ → ℝ}
    (b_pos : ∀ n, 0 < b n) :
    ∃ v : ℕ → E,
      Tendsto (fun n => ∏ i in range (n + 1), v i) atTop (𝓝 a) ∧
        (∀ n, v n ∈ s) ∧ ‖v 0 / a‖ < b 0 ∧ ∀ n, 0 < n → ‖v n‖ < b n := by
  obtain ⟨u : ℕ → E, u_in : ∀ n, u n ∈ s, lim_u : Tendsto u atTop (𝓝 a)⟩ :=
    mem_closure_iff_seq_limit.mp hg
  obtain ⟨n₀, hn₀⟩ : ∃ n₀, ∀ n ≥ n₀, ‖u n / a‖ < b 0 :=
    haveI : { x | ‖x / a‖ < b 0 } ∈ 𝓝 a := by
      simp_rw [← dist_eq_norm_div]
      exact Metric.ball_mem_nhds _ (b_pos _)
    Filter.tendsto_atTop'.mp lim_u _ this
  set z : ℕ → E := fun n => u (n + n₀)
  have lim_z : Tendsto z atTop (𝓝 a) := lim_u.comp (tendsto_add_atTop_nat n₀)
  have mem_𝓤 : ∀ n, { p : E × E | ‖p.1 / p.2‖ < b (n + 1) } ∈ 𝓤 E := fun n => by
    simpa [← dist_eq_norm_div] using Metric.dist_mem_uniformity (b_pos <| n + 1)
  obtain ⟨φ : ℕ → ℕ, φ_extr : StrictMono φ, hφ : ∀ n, ‖z (φ <| n + 1) / z (φ n)‖ < b (n + 1)⟩ :=
    lim_z.cauchySeq.subseq_mem mem_𝓤
  set w : ℕ → E := z ∘ φ
  have hw : Tendsto w atTop (𝓝 a) := lim_z.comp φ_extr.tendsto_atTop
  set v : ℕ → E := fun i => if i = 0 then w 0 else w i / w (i - 1)
  refine' ⟨v, Tendsto.congr (Finset.eq_prod_range_div' w) hw, _, hn₀ _ (n₀.le_add_left _), _⟩
  · rintro ⟨⟩
    · change w 0 ∈ s
      apply u_in
    · apply s.div_mem <;> 

Proof state:
case intro.intro.intro.intro.intro.refine'_1.succ.hy
𝓕 : Type u_1
𝕜 : Type u_2
α : Type u_3
ι : Type u_4
κ : Type u_5
E : Type u_6
F : Type u_7
G : Type u_8
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
a a₁ a₂ b✝ b₁ b₂ : E
r r₁ r₂ : ℝ
s : Subgroup E
hg : a ∈ closure ↑s
b : ℕ → ℝ
b_pos : ∀ (n : ℕ), 0 < b n
u : ℕ → E
u_in : ∀ (n : ℕ), u n ∈ s
lim_u : Tendsto u atTop (𝓝 a)
n₀ : ℕ
hn₀ : ∀ n ≥ n₀, ‖u n / a‖ < b 0
z : ℕ → E := fun n => u (n + n₀)
lim_z : Tendsto z atTop (𝓝 a)
mem_𝓤 : ∀ (n : ℕ), {p | ‖p.1 / p.2‖ < b (n + 1)} ∈ 𝓤 E
φ : ℕ → ℕ
φ_extr : StrictMono φ
hφ : ∀ (n : ℕ), ‖z (φ (n + 1)) / z (φ n)‖ < b (n + 1)
w : ℕ → E := z ∘ φ
hw : Tendsto w atTop (𝓝 a)
v : ℕ → E := fun i => if i = 0 then w 0 else w i / w (i - 1)
n✝ : ℕ
⊢ w (Nat.succ n✝ - 1) ∈ s

Next tactic:
apply u_in
@[to_additive]
theorem controlled_prod_of_mem_closure {s : Subgroup E} (hg : a ∈ closure (s : Set E)) {b : ℕ → ℝ}
    (b_pos : ∀ n, 0 < b n) :
    ∃ v : ℕ → E,
      Tendsto (fun n => ∏ i in range (n + 1), v i) atTop (𝓝 a) ∧
        (∀ n, v n ∈ s) ∧ ‖v 0 / a‖ < b 0 ∧ ∀ n, 0 < n → ‖v n‖ < b n := by
  obtain ⟨u : ℕ → E, u_in : ∀ n, u n ∈ s, lim_u : Tendsto u atTop (𝓝 a)⟩ :=
    mem_closure_iff_seq_limit.mp hg
  obtain ⟨n₀, hn₀⟩ : ∃ n₀, ∀ n ≥ n₀, ‖u n / a‖ < b 0 :=
    haveI : { x | ‖x / a‖ < b 0 } ∈ 𝓝 a := by
      simp_rw [← dist_eq_norm_div]
      exact Metric.ball_mem_nhds _ (b_pos _)
    Filter.tendsto_atTop'.mp lim_u _ this
  set z : ℕ → E := fun n => u (n + n₀)
  have lim_z : Tendsto z atTop (𝓝 a) := lim_u.comp (tendsto_add_atTop_nat n₀)
  have mem_𝓤 : ∀ n, { p : E × E | ‖p.1 / p.2‖ < b (n + 1) } ∈ 𝓤 E := fun n => by
    simpa [← dist_eq_norm_div] using Metric.dist_mem_uniformity (b_pos <| n + 1)
  obtain ⟨φ : ℕ → ℕ, φ_extr : StrictMono φ, hφ : ∀ n, ‖z (φ <| n + 1) / z (φ n)‖ < b (n + 1)⟩ :=
    lim_z.cauchySeq.subseq_mem mem_𝓤
  set w : ℕ → E := z ∘ φ
  have hw : Tendsto w atTop (𝓝 a) := lim_z.comp φ_extr.tendsto_atTop
  set v : ℕ → E := fun i => if i = 0 then w 0 else w i / w (i - 1)
  refine' ⟨v, Tendsto.congr (Finset.eq_prod_range_div' w) hw, _, hn₀ _ (n₀.le_add_left _), _⟩
  · rintro ⟨⟩
    · change w 0 ∈ s
      apply u_in
    · apply s.div_mem <;> apply u_in
  · 

Proof state:
case intro.intro.intro.intro.intro.refine'_2
𝓕 : Type u_1
𝕜 : Type u_2
α : Type u_3
ι : Type u_4
κ : Type u_5
E : Type u_6
F : Type u_7
G : Type u_8
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
a a₁ a₂ b✝ b₁ b₂ : E
r r₁ r₂ : ℝ
s : Subgroup E
hg : a ∈ closure ↑s
b : ℕ → ℝ
b_pos : ∀ (n : ℕ), 0 < b n
u : ℕ → E
u_in : ∀ (n : ℕ), u n ∈ s
lim_u : Tendsto u atTop (𝓝 a)
n₀ : ℕ
hn₀ : ∀ n ≥ n₀, ‖u n / a‖ < b 0
z : ℕ → E := fun n => u (n + n₀)
lim_z : Tendsto z atTop (𝓝 a)
mem_𝓤 : ∀ (n : ℕ), {p | ‖p.1 / p.2‖ < b (n + 1)} ∈ 𝓤 E
φ : ℕ → ℕ
φ_extr : StrictMono φ
hφ : ∀ (n : ℕ), ‖z (φ (n + 1)) / z (φ n)‖ < b (n + 1)
w : ℕ → E := z ∘ φ
hw : Tendsto w atTop (𝓝 a)
v : ℕ → E := fun i => if i = 0 then w 0 else w i / w (i - 1)
⊢ ∀ (n : ℕ), 0 < n → ‖v n‖ < b n

Next tactic:
intro l hl
@[to_additive]
theorem controlled_prod_of_mem_closure {s : Subgroup E} (hg : a ∈ closure (s : Set E)) {b : ℕ → ℝ}
    (b_pos : ∀ n, 0 < b n) :
    ∃ v : ℕ → E,
      Tendsto (fun n => ∏ i in range (n + 1), v i) atTop (𝓝 a) ∧
        (∀ n, v n ∈ s) ∧ ‖v 0 / a‖ < b 0 ∧ ∀ n, 0 < n → ‖v n‖ < b n := by
  obtain ⟨u : ℕ → E, u_in : ∀ n, u n ∈ s, lim_u : Tendsto u atTop (𝓝 a)⟩ :=
    mem_closure_iff_seq_limit.mp hg
  obtain ⟨n₀, hn₀⟩ : ∃ n₀, ∀ n ≥ n₀, ‖u n / a‖ < b 0 :=
    haveI : { x | ‖x / a‖ < b 0 } ∈ 𝓝 a := by
      simp_rw [← dist_eq_norm_div]
      exact Metric.ball_mem_nhds _ (b_pos _)
    Filter.tendsto_atTop'.mp lim_u _ this
  set z : ℕ → E := fun n => u (n + n₀)
  have lim_z : Tendsto z atTop (𝓝 a) := lim_u.comp (tendsto_add_atTop_nat n₀)
  have mem_𝓤 : ∀ n, { p : E × E | ‖p.1 / p.2‖ < b (n + 1) } ∈ 𝓤 E := fun n => by
    simpa [← dist_eq_norm_div] using Metric.dist_mem_uniformity (b_pos <| n + 1)
  obtain ⟨φ : ℕ → ℕ, φ_extr : StrictMono φ, hφ : ∀ n, ‖z (φ <| n + 1) / z (φ n)‖ < b (n + 1)⟩ :=
    lim_z.cauchySeq.subseq_mem mem_𝓤
  set w : ℕ → E := z ∘ φ
  have hw : Tendsto w atTop (𝓝 a) := lim_z.comp φ_extr.tendsto_atTop
  set v : ℕ → E := fun i => if i = 0 then w 0 else w i / w (i - 1)
  refine' ⟨v, Tendsto.congr (Finset.eq_prod_range_div' w) hw, _, hn₀ _ (n₀.le_add_left _), _⟩
  · rintro ⟨⟩
    · change w 0 ∈ s
      apply u_in
    · apply s.div_mem <;> apply u_in
  · intro l hl
    

Proof state:
case intro.intro.intro.intro.intro.refine'_2
𝓕 : Type u_1
𝕜 : Type u_2
α : Type u_3
ι : Type u_4
κ : Type u_5
E : Type u_6
F : Type u_7
G : Type u_8
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
a a₁ a₂ b✝ b₁ b₂ : E
r r₁ r₂ : ℝ
s : Subgroup E
hg : a ∈ closure ↑s
b : ℕ → ℝ
b_pos : ∀ (n : ℕ), 0 < b n
u : ℕ → E
u_in : ∀ (n : ℕ), u n ∈ s
lim_u : Tendsto u atTop (𝓝 a)
n₀ : ℕ
hn₀ : ∀ n ≥ n₀, ‖u n / a‖ < b 0
z : ℕ → E := fun n => u (n + n₀)
lim_z : Tendsto z atTop (𝓝 a)
mem_𝓤 : ∀ (n : ℕ), {p | ‖p.1 / p.2‖ < b (n + 1)} ∈ 𝓤 E
φ : ℕ → ℕ
φ_extr : StrictMono φ
hφ : ∀ (n : ℕ), ‖z (φ (n + 1)) / z (φ n)‖ < b (n + 1)
w : ℕ → E := z ∘ φ
hw : Tendsto w atTop (𝓝 a)
v : ℕ → E := fun i => if i = 0 then w 0 else w i / w (i - 1)
l : ℕ
hl : 0 < l
⊢ ‖v l‖ < b l

Next tactic:
obtain ⟨k, rfl⟩ : ∃ k, l = k + 1
@[to_additive]
theorem controlled_prod_of_mem_closure {s : Subgroup E} (hg : a ∈ closure (s : Set E)) {b : ℕ → ℝ}
    (b_pos : ∀ n, 0 < b n) :
    ∃ v : ℕ → E,
      Tendsto (fun n => ∏ i in range (n + 1), v i) atTop (𝓝 a) ∧
        (∀ n, v n ∈ s) ∧ ‖v 0 / a‖ < b 0 ∧ ∀ n, 0 < n → ‖v n‖ < b n := by
  obtain ⟨u : ℕ → E, u_in : ∀ n, u n ∈ s, lim_u : Tendsto u atTop (𝓝 a)⟩ :=
    mem_closure_iff_seq_limit.mp hg
  obtain ⟨n₀, hn₀⟩ : ∃ n₀, ∀ n ≥ n₀, ‖u n / a‖ < b 0 :=
    haveI : { x | ‖x / a‖ < b 0 } ∈ 𝓝 a := by
      simp_rw [← dist_eq_norm_div]
      exact Metric.ball_mem_nhds _ (b_pos _)
    Filter.tendsto_atTop'.mp lim_u _ this
  set z : ℕ → E := fun n => u (n + n₀)
  have lim_z : Tendsto z atTop (𝓝 a) := lim_u.comp (tendsto_add_atTop_nat n₀)
  have mem_𝓤 : ∀ n, { p : E × E | ‖p.1 / p.2‖ < b (n + 1) } ∈ 𝓤 E := fun n => by
    simpa [← dist_eq_norm_div] using Metric.dist_mem_uniformity (b_pos <| n + 1)
  obtain ⟨φ : ℕ → ℕ, φ_extr : StrictMono φ, hφ : ∀ n, ‖z (φ <| n + 1) / z (φ n)‖ < b (n + 1)⟩ :=
    lim_z.cauchySeq.subseq_mem mem_𝓤
  set w : ℕ → E := z ∘ φ
  have hw : Tendsto w atTop (𝓝 a) := lim_z.comp φ_extr.tendsto_atTop
  set v : ℕ → E := fun i => if i = 0 then w 0 else w i / w (i - 1)
  refine' ⟨v, Tendsto.congr (Finset.eq_prod_range_div' w) hw, _, hn₀ _ (n₀.le_add_left _), _⟩
  · rintro ⟨⟩
    · change w 0 ∈ s
      apply u_in
    · apply s.div_mem <;> apply u_in
  · intro l hl
    obtain ⟨k, rfl⟩ : ∃ k, l = k + 1
    

Proof state:
𝓕 : Type u_1
𝕜 : Type u_2
α : Type u_3
ι : Type u_4
κ : Type u_5
E : Type u_6
F : Type u_7
G : Type u_8
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
a a₁ a₂ b✝ b₁ b₂ : E
r r₁ r₂ : ℝ
s : Subgroup E
hg : a ∈ closure ↑s
b : ℕ → ℝ
b_pos : ∀ (n : ℕ), 0 < b n
u : ℕ → E
u_in : ∀ (n : ℕ), u n ∈ s
lim_u : Tendsto u atTop (𝓝 a)
n₀ : ℕ
hn₀ : ∀ n ≥ n₀, ‖u n / a‖ < b 0
z : ℕ → E := fun n => u (n + n₀)
lim_z : Tendsto z atTop (𝓝 a)
mem_𝓤 : ∀ (n : ℕ), {p | ‖p.1 / p.2‖ < b (n + 1)} ∈ 𝓤 E
φ : ℕ → ℕ
φ_extr : StrictMono φ
hφ : ∀ (n : ℕ), ‖z (φ (n + 1)) / z (φ n)‖ < b (n + 1)
w : ℕ → E := z ∘ φ
hw : Tendsto w atTop (𝓝 a)
v : ℕ → E := fun i => if i = 0 then w 0 else w i / w (i - 1)
l : ℕ
hl : 0 < l
⊢ ∃ k, l = k + 1
case intro.intro.intro.intro.intro.refine'_2.intro
𝓕 : Type u_1
𝕜 : Type u_2
α : Type u_3
ι : Type u_4
κ : Type u_5
E : Type u_6
F : Type u_7
G : Type u_8
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
a a₁ a₂ b✝ b₁ b₂ : E
r r₁ r₂ : ℝ
s : Subgroup E
hg : a ∈ closure ↑s
b : ℕ → ℝ
b_pos : ∀ (n : ℕ), 0 < b n
u : ℕ → E
u_in : ∀ (n : ℕ), u n ∈ s
lim_u : Tendsto u atTop (𝓝 a)
n₀ : ℕ
hn₀ : ∀ n ≥ n₀, ‖u n / a‖ < b 0
z : ℕ → E := fun n => u (n + n₀)
lim_z : Tendsto z atTop (𝓝 a)
mem_𝓤 : ∀ (n : ℕ), {p | ‖p.1 / p.2‖ < b (n + 1)} ∈ 𝓤 E
φ : ℕ → ℕ
φ_extr : StrictMono φ
hφ : ∀ (n : ℕ), ‖z (φ (n + 1)) / z (φ n)‖ < b (n + 1)
w : ℕ → E := z ∘ φ
hw : Tendsto w atTop (𝓝 a)
v : ℕ → E := fun i => if i = 0 then w 0 else w i / w (i - 1)
k : ℕ
hl : 0 < k + 1
⊢ ‖v (k + 1)‖ < b (k + 1)

Next tactic:
exact Nat.exists_eq_succ_of_ne_zero hl.ne'
@[to_additive]
theorem controlled_prod_of_mem_closure {s : Subgroup E} (hg : a ∈ closure (s : Set E)) {b : ℕ → ℝ}
    (b_pos : ∀ n, 0 < b n) :
    ∃ v : ℕ → E,
      Tendsto (fun n => ∏ i in range (n + 1), v i) atTop (𝓝 a) ∧
        (∀ n, v n ∈ s) ∧ ‖v 0 / a‖ < b 0 ∧ ∀ n, 0 < n → ‖v n‖ < b n := by
  obtain ⟨u : ℕ → E, u_in : ∀ n, u n ∈ s, lim_u : Tendsto u atTop (𝓝 a)⟩ :=
    mem_closure_iff_seq_limit.mp hg
  obtain ⟨n₀, hn₀⟩ : ∃ n₀, ∀ n ≥ n₀, ‖u n / a‖ < b 0 :=
    haveI : { x | ‖x / a‖ < b 0 } ∈ 𝓝 a := by
      simp_rw [← dist_eq_norm_div]
      exact Metric.ball_mem_nhds _ (b_pos _)
    Filter.tendsto_atTop'.mp lim_u _ this
  set z : ℕ → E := fun n => u (n + n₀)
  have lim_z : Tendsto z atTop (𝓝 a) := lim_u.comp (tendsto_add_atTop_nat n₀)
  have mem_𝓤 : ∀ n, { p : E × E | ‖p.1 / p.2‖ < b (n + 1) } ∈ 𝓤 E := fun n => by
    simpa [← dist_eq_norm_div] using Metric.dist_mem_uniformity (b_pos <| n + 1)
  obtain ⟨φ : ℕ → ℕ, φ_extr : StrictMono φ, hφ : ∀ n, ‖z (φ <| n + 1) / z (φ n)‖ < b (n + 1)⟩ :=
    lim_z.cauchySeq.subseq_mem mem_𝓤
  set w : ℕ → E := z ∘ φ
  have hw : Tendsto w atTop (𝓝 a) := lim_z.comp φ_extr.tendsto_atTop
  set v : ℕ → E := fun i => if i = 0 then w 0 else w i / w (i - 1)
  refine' ⟨v, Tendsto.congr (Finset.eq_prod_range_div' w) hw, _, hn₀ _ (n₀.le_add_left _), _⟩
  · rintro ⟨⟩
    · change w 0 ∈ s
      apply u_in
    · apply s.div_mem <;> apply u_in
  · intro l hl
    obtain ⟨k, rfl⟩ : ∃ k, l = k + 1
    exact Nat.exists_eq_succ_of_ne_zero hl.ne'
    

Proof state:
case intro.intro.intro.intro.intro.refine'_2.intro
𝓕 : Type u_1
𝕜 : Type u_2
α : Type u_3
ι : Type u_4
κ : Type u_5
E : Type u_6
F : Type u_7
G : Type u_8
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
a a₁ a₂ b✝ b₁ b₂ : E
r r₁ r₂ : ℝ
s : Subgroup E
hg : a ∈ closure ↑s
b : ℕ → ℝ
b_pos : ∀ (n : ℕ), 0 < b n
u : ℕ → E
u_in : ∀ (n : ℕ), u n ∈ s
lim_u : Tendsto u atTop (𝓝 a)
n₀ : ℕ
hn₀ : ∀ n ≥ n₀, ‖u n / a‖ < b 0
z : ℕ → E := fun n => u (n + n₀)
lim_z : Tendsto z atTop (𝓝 a)
mem_𝓤 : ∀ (n : ℕ), {p | ‖p.1 / p.2‖ < b (n + 1)} ∈ 𝓤 E
φ : ℕ → ℕ
φ_extr : StrictMono φ
hφ : ∀ (n : ℕ), ‖z (φ (n + 1)) / z (φ n)‖ < b (n + 1)
w : ℕ → E := z ∘ φ
hw : Tendsto w atTop (𝓝 a)
v : ℕ → E := fun i => if i = 0 then w 0 else w i / w (i - 1)
k : ℕ
hl : 0 < k + 1
⊢ ‖v (k + 1)‖ < b (k + 1)

Next tactic:
apply hφ
@[to_additive]
theorem controlled_prod_of_mem_closure_range {j : E →* F} {b : F}
    (hb : b ∈ closure (j.range : Set F)) {f : ℕ → ℝ} (b_pos : ∀ n, 0 < f n) :
    ∃ a : ℕ → E,
      Tendsto (fun n => ∏ i in range (n + 1), j (a i)) atTop (𝓝 b) ∧
        ‖j (a 0) / b‖ < f 0 ∧ ∀ n, 0 < n → ‖j (a n)‖ < f n := by
  

Proof state:
𝓕 : Type u_1
𝕜 : Type u_2
α : Type u_3
ι : Type u_4
κ : Type u_5
E : Type u_6
F : Type u_7
G : Type u_8
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
a a₁ a₂ b✝ b₁ b₂ : E
r r₁ r₂ : ℝ
j : E →* F
b : F
hb : b ∈ closure ↑(MonoidHom.range j)
f : ℕ → ℝ
b_pos : ∀ (n : ℕ), 0 < f n
⊢ ∃ a,
    Tendsto (fun n => ∏ i in range (n + 1), j (a i)) atTop (𝓝 b) ∧
      ‖j (a 0) / b‖ < f 0 ∧ ∀ (n : ℕ), 0 < n → ‖j (a n)‖ < f n

Next tactic:
obtain ⟨v, sum_v, v_in, hv₀, hv_pos⟩ := controlled_prod_of_mem_closure hb b_pos
@[to_additive]
theorem controlled_prod_of_mem_closure_range {j : E →* F} {b : F}
    (hb : b ∈ closure (j.range : Set F)) {f : ℕ → ℝ} (b_pos : ∀ n, 0 < f n) :
    ∃ a : ℕ → E,
      Tendsto (fun n => ∏ i in range (n + 1), j (a i)) atTop (𝓝 b) ∧
        ‖j (a 0) / b‖ < f 0 ∧ ∀ n, 0 < n → ‖j (a n)‖ < f n := by
  obtain ⟨v, sum_v, v_in, hv₀, hv_pos⟩ := controlled_prod_of_mem_closure hb b_pos
  

Proof state:
case intro.intro.intro.intro
𝓕 : Type u_1
𝕜 : Type u_2
α : Type u_3
ι : Type u_4
κ : Type u_5
E : Type u_6
F : Type u_7
G : Type u_8
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
a a₁ a₂ b✝ b₁ b₂ : E
r r₁ r₂ : ℝ
j : E →* F
b : F
hb : b ∈ closure ↑(MonoidHom.range j)
f : ℕ → ℝ
b_pos : ∀ (n : ℕ), 0 < f n
v : ℕ → F
sum_v : Tendsto (fun n => ∏ i in range (n + 1), v i) atTop (𝓝 b)
v_in : ∀ (n : ℕ), v n ∈ MonoidHom.range j
hv₀ : ‖v 0 / b‖ < f 0
hv_pos : ∀ (n : ℕ), 0 < n → ‖v n‖ < f n
⊢ ∃ a,
    Tendsto (fun n => ∏ i in range (n + 1), j (a i)) atTop (𝓝 b) ∧
      ‖j (a 0) / b‖ < f 0 ∧ ∀ (n : ℕ), 0 < n → ‖j (a n)‖ < f n

Next tactic:
choose g hg using v_in
@[to_additive]
theorem controlled_prod_of_mem_closure_range {j : E →* F} {b : F}
    (hb : b ∈ closure (j.range : Set F)) {f : ℕ → ℝ} (b_pos : ∀ n, 0 < f n) :
    ∃ a : ℕ → E,
      Tendsto (fun n => ∏ i in range (n + 1), j (a i)) atTop (𝓝 b) ∧
        ‖j (a 0) / b‖ < f 0 ∧ ∀ n, 0 < n → ‖j (a n)‖ < f n := by
  obtain ⟨v, sum_v, v_in, hv₀, hv_pos⟩ := controlled_prod_of_mem_closure hb b_pos
  choose g hg using v_in
  

Proof state:
case intro.intro.intro.intro
𝓕 : Type u_1
𝕜 : Type u_2
α : Type u_3
ι : Type u_4
κ : Type u_5
E : Type u_6
F : Type u_7
G : Type u_8
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
a a₁ a₂ b✝ b₁ b₂ : E
r r₁ r₂ : ℝ
j : E →* F
b : F
hb : b ∈ closure ↑(MonoidHom.range j)
f : ℕ → ℝ
b_pos : ∀ (n : ℕ), 0 < f n
v : ℕ → F
sum_v : Tendsto (fun n => ∏ i in range (n + 1), v i) atTop (𝓝 b)
hv₀ : ‖v 0 / b‖ < f 0
hv_pos : ∀ (n : ℕ), 0 < n → ‖v n‖ < f n
g : ℕ → E
hg : ∀ (n : ℕ), j (g n) = v n
⊢ ∃ a,
    Tendsto (fun n => ∏ i in range (n + 1), j (a i)) atTop (𝓝 b) ∧
      ‖j (a 0) / b‖ < f 0 ∧ ∀ (n : ℕ), 0 < n → ‖j (a n)‖ < f n

Next tactic:
exact
    ⟨g, by simpa [← hg] using sum_v, by simpa [hg 0] using hv₀,
      fun n hn => by simpa [hg] using hv_pos n hn⟩
@[to_additive]
theorem controlled_prod_of_mem_closure_range {j : E →* F} {b : F}
    (hb : b ∈ closure (j.range : Set F)) {f : ℕ → ℝ} (b_pos : ∀ n, 0 < f n) :
    ∃ a : ℕ → E,
      Tendsto (fun n => ∏ i in range (n + 1), j (a i)) atTop (𝓝 b) ∧
        ‖j (a 0) / b‖ < f 0 ∧ ∀ n, 0 < n → ‖j (a n)‖ < f n := by
  obtain ⟨v, sum_v, v_in, hv₀, hv_pos⟩ := controlled_prod_of_mem_closure hb b_pos
  choose g hg using v_in
  exact
    ⟨g, by 

Proof state:
𝓕 : Type u_1
𝕜 : Type u_2
α : Type u_3
ι : Type u_4
κ : Type u_5
E : Type u_6
F : Type u_7
G : Type u_8
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
a a₁ a₂ b✝ b₁ b₂ : E
r r₁ r₂ : ℝ
j : E →* F
b : F
hb : b ∈ closure ↑(MonoidHom.range j)
f : ℕ → ℝ
b_pos : ∀ (n : ℕ), 0 < f n
v : ℕ → F
sum_v : Tendsto (fun n => ∏ i in range (n + 1), v i) atTop (𝓝 b)
hv₀ : ‖v 0 / b‖ < f 0
hv_pos : ∀ (n : ℕ), 0 < n → ‖v n‖ < f n
g : ℕ → E
hg : ∀ (n : ℕ), j (g n) = v n
⊢ Tendsto (fun n => ∏ i in range (n + 1), j (g i)) atTop (𝓝 b)

Next tactic:
simpa [← hg] using sum_v
@[to_additive]
theorem controlled_prod_of_mem_closure_range {j : E →* F} {b : F}
    (hb : b ∈ closure (j.range : Set F)) {f : ℕ → ℝ} (b_pos : ∀ n, 0 < f n) :
    ∃ a : ℕ → E,
      Tendsto (fun n => ∏ i in range (n + 1), j (a i)) atTop (𝓝 b) ∧
        ‖j (a 0) / b‖ < f 0 ∧ ∀ n, 0 < n → ‖j (a n)‖ < f n := by
  obtain ⟨v, sum_v, v_in, hv₀, hv_pos⟩ := controlled_prod_of_mem_closure hb b_pos
  choose g hg using v_in
  exact
    ⟨g, by simpa [← hg] using sum_v, by 

Proof state:
𝓕 : Type u_1
𝕜 : Type u_2
α : Type u_3
ι : Type u_4
κ : Type u_5
E : Type u_6
F : Type u_7
G : Type u_8
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
a a₁ a₂ b✝ b₁ b₂ : E
r r₁ r₂ : ℝ
j : E →* F
b : F
hb : b ∈ closure ↑(MonoidHom.range j)
f : ℕ → ℝ
b_pos : ∀ (n : ℕ), 0 < f n
v : ℕ → F
sum_v : Tendsto (fun n => ∏ i in range (n + 1), v i) atTop (𝓝 b)
hv₀ : ‖v 0 / b‖ < f 0
hv_pos : ∀ (n : ℕ), 0 < n → ‖v n‖ < f n
g : ℕ → E
hg : ∀ (n : ℕ), j (g n) = v n
⊢ ‖j (g 0) / b‖ < f 0

Next tactic:
simpa [hg 0] using hv₀