-- Mathlib/Analysis/Normed/Module/FiniteDimension.lean
/-- A locally compact normed vector space is proper. -/
lemma ProperSpace.of_locallyCompactSpace (𝕜 : Type*) [NontriviallyNormedField 𝕜]
    {E : Type*} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] [LocallyCompactSpace E] :
    ProperSpace E

Proof state:
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : LocallyCompactSpace E
r : ℝ
rpos : 0 < r
hr : IsCompact (closedBall 0 r)
c : 𝕜
hc : 1 < ‖c‖
n : ℕ
this : c ^ n ≠ 0
⊢ IsCompact (closedBall 0 (‖c‖ ^ n * r))

Tactic:
simpa [_root_.smul_closedBall' this] using hr.smul (c ^ n)

Result:
no goals

Proof state:
case intro.intro
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : LocallyCompactSpace E
r : ℝ
rpos : 0 < r
hr : IsCompact (closedBall 0 r)
⊢ ProperSpace E

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

Result:
case intro.intro.intro
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : LocallyCompactSpace E
r : ℝ
rpos : 0 < r
hr : IsCompact (closedBall 0 r)
c : 𝕜
hc : 1 < ‖c‖
⊢ ProperSpace E

Proof state:
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : LocallyCompactSpace E
⊢ ProperSpace E

Tactic:
rcases exists_isCompact_closedBall (0 : E) with ⟨r, rpos, hr⟩

Result:
case intro.intro
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : LocallyCompactSpace E
r : ℝ
rpos : 0 < r
hr : IsCompact (closedBall 0 r)
⊢ ProperSpace E

Proof state:
case intro.intro.intro
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : LocallyCompactSpace E
r : ℝ
rpos : 0 < r
hr : IsCompact (closedBall 0 r)
c : 𝕜
hc : 1 < ‖c‖
⊢ ProperSpace E

Tactic:
have hC : ∀ n, IsCompact (closedBall (0 : E) (‖c‖^n * r)) := fun n ↦ by
  have : c ^ n ≠ 0 := pow_ne_zero _ <| fun h ↦ by simp [h, zero_le_one.not_lt] at hc
  simpa [_root_.smul_closedBall' this] using hr.smul (c ^ n)

Result:
case intro.intro.intro
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : LocallyCompactSpace E
r : ℝ
rpos : 0 < r
hr : IsCompact (closedBall 0 r)
c : 𝕜
hc : 1 < ‖c‖
hC : ∀ (n : ℕ), IsCompact (closedBall 0 (‖c‖ ^ n * r))
⊢ ProperSpace E

Proof state:
case intro.intro.intro
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : LocallyCompactSpace E
r : ℝ
rpos : 0 < r
hr : IsCompact (closedBall 0 r)
c : 𝕜
hc : 1 < ‖c‖
hC : ∀ (n : ℕ), IsCompact (closedBall 0 (‖c‖ ^ n * r))
hTop : Tendsto (fun n => ‖c‖ ^ n * r) atTop atTop
⊢ ProperSpace E

Tactic:
exact .of_seq_closedBall hTop (Eventually.of_forall hC)

Result:
no goals

Proof state:
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : LocallyCompactSpace E
r : ℝ
rpos : 0 < r
hr : IsCompact (closedBall 0 r)
c : 𝕜
hc : 1 < ‖c‖
n : ℕ
h : c = 0
⊢ False

Tactic:
simp [h, zero_le_one.not_lt] at hc

Result:
no goals

Proof state:
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : LocallyCompactSpace E
r : ℝ
rpos : 0 < r
hr : IsCompact (closedBall 0 r)
c : 𝕜
hc : 1 < ‖c‖
n : ℕ
⊢ IsCompact (closedBall 0 (‖c‖ ^ n * r))

Tactic:
have : c ^ n ≠ 0 := pow_ne_zero _ <| fun h ↦ by simp [h, zero_le_one.not_lt] at hc

Result:
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : LocallyCompactSpace E
r : ℝ
rpos : 0 < r
hr : IsCompact (closedBall 0 r)
c : 𝕜
hc : 1 < ‖c‖
n : ℕ
this : c ^ n ≠ 0
⊢ IsCompact (closedBall 0 (‖c‖ ^ n * r))

Proof state:
case intro.intro.intro
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : LocallyCompactSpace E
r : ℝ
rpos : 0 < r
hr : IsCompact (closedBall 0 r)
c : 𝕜
hc : 1 < ‖c‖
hC : ∀ (n : ℕ), IsCompact (closedBall 0 (‖c‖ ^ n * r))
⊢ ProperSpace E

Tactic:
have hTop : Tendsto (fun n ↦ ‖c‖^n * r) atTop atTop :=
  Tendsto.atTop_mul_const rpos (tendsto_pow_atTop_atTop_of_one_lt hc)

Result:
case intro.intro.intro
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : LocallyCompactSpace E
r : ℝ
rpos : 0 < r
hr : IsCompact (closedBall 0 r)
c : 𝕜
hc : 1 < ‖c‖
hC : ∀ (n : ℕ), IsCompact (closedBall 0 (‖c‖ ^ n * r))
hTop : Tendsto (fun n => ‖c‖ ^ n * r) atTop atTop
⊢ ProperSpace E