-- Mathlib/Analysis/Normed/Module/FiniteDimension.lean
theorem LinearIndependent.eventually {ι} [Finite ι] {f : ι → E}
    (hf : LinearIndependent 𝕜 f) : ∀ᶠ g in 𝓝 f, LinearIndependent 𝕜 g

Proof state:
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type v
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CompleteSpace 𝕜
ι : Type u_1
inst✝ : Finite ι
f : ι → E
val✝ : Fintype ι
hf : LinearMap.ker ((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (f i)) = ⊥
K : ℝ≥0
K0 : K > 0
hK : AntilipschitzWith K ⇑((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (f i))
this : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)
g : ι → E
hg : ∑ i : ι, ‖g i - f i‖ < ((fun a => ↑a) K)⁻¹
⊢ ↑(∑ i : ι, ‖g i - f i‖₊) < ↑K⁻¹

Tactic:
push_cast

Result:
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type v
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CompleteSpace 𝕜
ι : Type u_1
inst✝ : Finite ι
f : ι → E
val✝ : Fintype ι
hf : LinearMap.ker ((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (f i)) = ⊥
K : ℝ≥0
K0 : K > 0
hK : AntilipschitzWith K ⇑((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (f i))
this : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)
g : ι → E
hg : ∑ i : ι, ‖g i - f i‖ < ((fun a => ↑a) K)⁻¹
⊢ ∑ x : ι, ‖g x - f x‖ < (↑K)⁻¹

Proof state:
case intro.intro.intro
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type v
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CompleteSpace 𝕜
ι : Type u_1
inst✝ : Finite ι
f : ι → E
val✝ : Fintype ι
hf : LinearMap.ker ((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (f i)) = ⊥
K : ℝ≥0
K0 : K > 0
hK : AntilipschitzWith K ⇑((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (f i))
this : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)
g : ι → E
hg : ∑ i : ι, ‖g i - f i‖₊ < K⁻¹
v u : ι → 𝕜
⊢ ‖∑ x : ι, (v x - u x) • (g x - f x)‖ ≤ ∑ i : ι, ‖g i - f i‖ * ‖v - u‖

Tactic:
refine norm_sum_le_of_le _ fun i _ => ?_

Result:
case intro.intro.intro
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type v
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CompleteSpace 𝕜
ι : Type u_1
inst✝ : Finite ι
f : ι → E
val✝ : Fintype ι
hf : LinearMap.ker ((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (f i)) = ⊥
K : ℝ≥0
K0 : K > 0
hK : AntilipschitzWith K ⇑((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (f i))
this : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)
g : ι → E
hg : ∑ i : ι, ‖g i - f i‖₊ < K⁻¹
v u : ι → 𝕜
i : ι
x✝ : i ∈ Finset.univ
⊢ ‖(v i - u i) • (g i - f i)‖ ≤ ‖g i - f i‖ * ‖v - u‖

Proof state:
case intro.intro.intro
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type v
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CompleteSpace 𝕜
ι : Type u_1
inst✝ : Finite ι
f : ι → E
val✝ : Fintype ι
hf : LinearMap.ker ((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (f i)) = ⊥
K : ℝ≥0
K0 : K > 0
hK : AntilipschitzWith K ⇑((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (f i))
this : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)
g : ι → E
hg : ∑ i : ι, ‖g i - f i‖₊ < K⁻¹
v u : ι → 𝕜
i : ι
x✝ : i ∈ Finset.univ
⊢ ‖(v i - u i) • (g i - f i)‖ ≤ ‖g i - f i‖ * ‖v - u‖

Tactic:
rw [norm_smul, mul_comm]

Result:
case intro.intro.intro
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type v
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CompleteSpace 𝕜
ι : Type u_1
inst✝ : Finite ι
f : ι → E
val✝ : Fintype ι
hf : LinearMap.ker ((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (f i)) = ⊥
K : ℝ≥0
K0 : K > 0
hK : AntilipschitzWith K ⇑((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (f i))
this : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)
g : ι → E
hg : ∑ i : ι, ‖g i - f i‖₊ < K⁻¹
v u : ι → 𝕜
i : ι
x✝ : i ∈ Finset.univ
⊢ ‖g i - f i‖ * ‖v i - u i‖ ≤ ‖g i - f i‖ * ‖v - u‖

Proof state:
case intro.intro.intro
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type v
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CompleteSpace 𝕜
ι : Type u_1
inst✝ : Finite ι
f : ι → E
val✝ : Fintype ι
hf : LinearMap.ker ((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (f i)) = ⊥
K : ℝ≥0
K0 : K > 0
hK : AntilipschitzWith K ⇑((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (f i))
this : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)
⊢ ∀ᶠ (g : ι → E) in 𝓝 f, LinearMap.ker ((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (g i)) = ⊥

Tactic:
refine (this.eventually (gt_mem_nhds <| inv_pos.2 K0)).mono fun g hg => ?_

Result:
case intro.intro.intro
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type v
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CompleteSpace 𝕜
ι : Type u_1
inst✝ : Finite ι
f : ι → E
val✝ : Fintype ι
hf : LinearMap.ker ((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (f i)) = ⊥
K : ℝ≥0
K0 : K > 0
hK : AntilipschitzWith K ⇑((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (f i))
this : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)
g : ι → E
hg : ∑ i : ι, ‖g i - f i‖ < ((fun a => ↑a) K)⁻¹
⊢ LinearMap.ker ((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (g i)) = ⊥

Proof state:
case intro.intro.intro
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type v
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CompleteSpace 𝕜
ι : Type u_1
inst✝ : Finite ι
f : ι → E
val✝ : Fintype ι
hf : LinearMap.ker ((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (f i)) = ⊥
K : ℝ≥0
K0 : K > 0
hK : AntilipschitzWith K ⇑((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (f i))
this : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)
g : ι → E
hg : ∑ i : ι, ‖g i - f i‖₊ < K⁻¹
⊢ LinearMap.ker ((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (g i)) = ⊥

Tactic:
rw [LinearMap.ker_eq_bot]

Result:
case intro.intro.intro
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type v
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CompleteSpace 𝕜
ι : Type u_1
inst✝ : Finite ι
f : ι → E
val✝ : Fintype ι
hf : LinearMap.ker ((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (f i)) = ⊥
K : ℝ≥0
K0 : K > 0
hK : AntilipschitzWith K ⇑((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (f i))
this : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)
g : ι → E
hg : ∑ i : ι, ‖g i - f i‖₊ < K⁻¹
⊢ Function.Injective ⇑((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (g i))