-- Mathlib/Topology/Algebra/Module/Cardinality.lean
theorem continuum_le_cardinal_of_nontriviallyNormedField
    (𝕜 : Type*) [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] : 𝔠 ≤ #𝕜

Proof state:
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
x : 𝕜
x✝ : x ∈ Set.univ
U : Set 𝕜
hU : U ∈ 𝓝 x
⊢ ∃ y ∈ U ∩ Set.univ, y ≠ x

Tactic:
rcases NormedField.exists_norm_lt_one 𝕜 with ⟨c, c_pos, hc⟩

Result:
case intro.intro
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
x : 𝕜
x✝ : x ∈ Set.univ
U : Set 𝕜
hU : U ∈ 𝓝 x
c : 𝕜
c_pos : 0 < ‖c‖
hc : ‖c‖ < 1
⊢ ∃ y ∈ U ∩ Set.univ, y ≠ x

Proof state:
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
x : 𝕜
x✝ : x ∈ Set.univ
U : Set 𝕜
hU : U ∈ 𝓝 x
c : 𝕜
c_pos : 0 < ‖c‖
hc : ‖c‖ < 1
A : Tendsto (fun n => x + c ^ n) atTop (𝓝 x)
B : ∀ᶠ (n : ℕ) in atTop, x + c ^ n ∈ U
n : ℕ
hn : x + c ^ n ∈ U
⊢ x + c ^ n ∈ U ∩ Set.univ

Tactic:
simpa using hn

Result:
no goals

Proof state:
case intro.intro.intro.h
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
x : 𝕜
x✝ : x ∈ Set.univ
U : Set 𝕜
hU : U ∈ 𝓝 x
c : 𝕜
c_pos : 0 < ‖c‖
hc : ‖c‖ < 1
A : Tendsto (fun n => x + c ^ n) atTop (𝓝 x)
B : ∀ᶠ (n : ℕ) in atTop, x + c ^ n ∈ U
n : ℕ
hn : x + c ^ n ∈ U
⊢ c ≠ 0

Tactic:
simpa using c_pos

Result:
no goals

Proof state:
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
⊢ ∃ f, range f ⊆ Set.univ ∧ Continuous f ∧ Injective f

Tactic:
apply Perfect.exists_nat_bool_injection _ univ_nonempty

Result:
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
⊢ Perfect Set.univ

Proof state:
case intro.intro.intro
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
f : (ℕ → Bool) → 𝕜
f_inj : Injective f
⊢ 𝔠 ≤ #𝕜

Tactic:
simpa using lift_mk_le_lift_mk_of_injective f_inj

Result:
no goals

Proof state:
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
this : ∃ f, range f ⊆ Set.univ ∧ Continuous f ∧ Injective f
⊢ 𝔠 ≤ #𝕜

Tactic:
rcases this with ⟨f, -, -, f_inj⟩

Result:
case intro.intro.intro
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
f : (ℕ → Bool) → 𝕜
f_inj : Injective f
⊢ 𝔠 ≤ #𝕜

Proof state:
case intro.intro.intro
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
x : 𝕜
x✝ : x ∈ Set.univ
U : Set 𝕜
hU : U ∈ 𝓝 x
c : 𝕜
c_pos : 0 < ‖c‖
hc : ‖c‖ < 1
A : Tendsto (fun n => x + c ^ n) atTop (𝓝 x)
B : ∀ᶠ (n : ℕ) in atTop, x + c ^ n ∈ U
n : ℕ
hn : x + c ^ n ∈ U
⊢ x + c ^ n ≠ x

Tactic:
simp only [ne_eq, add_right_eq_self]

Result:
case intro.intro.intro
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
x : 𝕜
x✝ : x ∈ Set.univ
U : Set 𝕜
hU : U ∈ 𝓝 x
c : 𝕜
c_pos : 0 < ‖c‖
hc : ‖c‖ < 1
A : Tendsto (fun n => x + c ^ n) atTop (𝓝 x)
B : ∀ᶠ (n : ℕ) in atTop, x + c ^ n ∈ U
n : ℕ
hn : x + c ^ n ∈ U
⊢ ¬c ^ n = 0

Proof state:
case intro.intro
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
x : 𝕜
x✝ : x ∈ Set.univ
U : Set 𝕜
hU : U ∈ 𝓝 x
c : 𝕜
c_pos : 0 < ‖c‖
hc : ‖c‖ < 1
⊢ ∃ y ∈ U ∩ Set.univ, y ≠ x

Tactic:
have A : Tendsto (fun n ↦ x + c^n) atTop (𝓝 (x + 0)) :=
  tendsto_const_nhds.add (tendsto_pow_atTop_nhds_zero_of_norm_lt_one hc)

Result:
case intro.intro
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
x : 𝕜
x✝ : x ∈ Set.univ
U : Set 𝕜
hU : U ∈ 𝓝 x
c : 𝕜
c_pos : 0 < ‖c‖
hc : ‖c‖ < 1
A : Tendsto (fun n => x + c ^ n) atTop (𝓝 (x + 0))
⊢ ∃ y ∈ U ∩ Set.univ, y ≠ x

Proof state:
case intro.intro
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
x : 𝕜
x✝ : x ∈ Set.univ
U : Set 𝕜
hU : U ∈ 𝓝 x
c : 𝕜
c_pos : 0 < ‖c‖
hc : ‖c‖ < 1
A : Tendsto (fun n => x + c ^ n) atTop (𝓝 (x + 0))
⊢ ∃ y ∈ U ∩ Set.univ, y ≠ x

Tactic:
rw [add_zero] at A

Result:
case intro.intro
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
x : 𝕜
x✝ : x ∈ Set.univ
U : Set 𝕜
hU : U ∈ 𝓝 x
c : 𝕜
c_pos : 0 < ‖c‖
hc : ‖c‖ < 1
A : Tendsto (fun n => x + c ^ n) atTop (𝓝 x)
⊢ ∃ y ∈ U ∩ Set.univ, y ≠ x

Proof state:
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
⊢ 𝔠 ≤ #𝕜

Tactic:
suffices ∃ f : (ℕ → Bool) → 𝕜, range f ⊆ univ ∧ Continuous f ∧ Injective f by
  rcases this with ⟨f, -, -, f_inj⟩
  simpa using lift_mk_le_lift_mk_of_injective f_inj

Result:
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
⊢ ∃ f, range f ⊆ Set.univ ∧ Continuous f ∧ Injective f

Proof state:
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
⊢ Perfect Set.univ

Tactic:
refine ⟨isClosed_univ, preperfect_iff_nhds.2 (fun x _ U hU ↦ ?_)⟩

Result:
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
x : 𝕜
x✝ : x ∈ Set.univ
U : Set 𝕜
hU : U ∈ 𝓝 x
⊢ ∃ y ∈ U ∩ Set.univ, y ≠ x

Proof state:
case intro.intro.intro
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
x : 𝕜
x✝ : x ∈ Set.univ
U : Set 𝕜
hU : U ∈ 𝓝 x
c : 𝕜
c_pos : 0 < ‖c‖
hc : ‖c‖ < 1
A : Tendsto (fun n => x + c ^ n) atTop (𝓝 x)
B : ∀ᶠ (n : ℕ) in atTop, x + c ^ n ∈ U
n : ℕ
hn : x + c ^ n ∈ U
⊢ ∃ y ∈ U ∩ Set.univ, y ≠ x

Tactic:
refine ⟨x + c^n, by simpa using hn, ?_⟩

Result:
case intro.intro.intro
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
x : 𝕜
x✝ : x ∈ Set.univ
U : Set 𝕜
hU : U ∈ 𝓝 x
c : 𝕜
c_pos : 0 < ‖c‖
hc : ‖c‖ < 1
A : Tendsto (fun n => x + c ^ n) atTop (𝓝 x)
B : ∀ᶠ (n : ℕ) in atTop, x + c ^ n ∈ U
n : ℕ
hn : x + c ^ n ∈ U
⊢ x + c ^ n ≠ x

Proof state:
case intro.intro
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
x : 𝕜
x✝ : x ∈ Set.univ
U : Set 𝕜
hU : U ∈ 𝓝 x
c : 𝕜
c_pos : 0 < ‖c‖
hc : ‖c‖ < 1
A : Tendsto (fun n => x + c ^ n) atTop (𝓝 x)
⊢ ∃ y ∈ U ∩ Set.univ, y ≠ x

Tactic:
have B : ∀ᶠ n in atTop, x + c^n ∈ U := tendsto_def.1 A U hU

Result:
case intro.intro
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
x : 𝕜
x✝ : x ∈ Set.univ
U : Set 𝕜
hU : U ∈ 𝓝 x
c : 𝕜
c_pos : 0 < ‖c‖
hc : ‖c‖ < 1
A : Tendsto (fun n => x + c ^ n) atTop (𝓝 x)
B : ∀ᶠ (n : ℕ) in atTop, x + c ^ n ∈ U
⊢ ∃ y ∈ U ∩ Set.univ, y ≠ x

Proof state:
case intro.intro.intro
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
x : 𝕜
x✝ : x ∈ Set.univ
U : Set 𝕜
hU : U ∈ 𝓝 x
c : 𝕜
c_pos : 0 < ‖c‖
hc : ‖c‖ < 1
A : Tendsto (fun n => x + c ^ n) atTop (𝓝 x)
B : ∀ᶠ (n : ℕ) in atTop, x + c ^ n ∈ U
n : ℕ
hn : x + c ^ n ∈ U
⊢ ¬c ^ n = 0

Tactic:
apply pow_ne_zero

Result:
case intro.intro.intro.h
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
x : 𝕜
x✝ : x ∈ Set.univ
U : Set 𝕜
hU : U ∈ 𝓝 x
c : 𝕜
c_pos : 0 < ‖c‖
hc : ‖c‖ < 1
A : Tendsto (fun n => x + c ^ n) atTop (𝓝 x)
B : ∀ᶠ (n : ℕ) in atTop, x + c ^ n ∈ U
n : ℕ
hn : x + c ^ n ∈ U
⊢ c ≠ 0

Proof state:
case intro.intro
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
x : 𝕜
x✝ : x ∈ Set.univ
U : Set 𝕜
hU : U ∈ 𝓝 x
c : 𝕜
c_pos : 0 < ‖c‖
hc : ‖c‖ < 1
A : Tendsto (fun n => x + c ^ n) atTop (𝓝 x)
B : ∀ᶠ (n : ℕ) in atTop, x + c ^ n ∈ U
⊢ ∃ y ∈ U ∩ Set.univ, y ≠ x

Tactic:
rcases B.exists with ⟨n, hn⟩

Result:
case intro.intro.intro
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
x : 𝕜
x✝ : x ∈ Set.univ
U : Set 𝕜
hU : U ∈ 𝓝 x
c : 𝕜
c_pos : 0 < ‖c‖
hc : ‖c‖ < 1
A : Tendsto (fun n => x + c ^ n) atTop (𝓝 x)
B : ∀ᶠ (n : ℕ) in atTop, x + c ^ n ∈ U
n : ℕ
hn : x + c ^ n ∈ U
⊢ ∃ y ∈ U ∩ Set.univ, y ≠ x