Theorem: Padic.exi_rat_seq_conv_cauchy
File: Mathlib/NumberTheory/Padics/PadicNumbers.lean

Proof steps:
state_before: p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
⊢ ∃ i, ∀ j ≥ i, padicNorm p (limSeq f j - limSeq f i) < ε
state_after: p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
⊢ ∃ i, ∀ j ≥ i, padicNorm p (limSeq f j - limSeq f i) < ε
tactic: have hε3 : 0 < ε / 3 := div_pos hε (by norm_num)
state_before: p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
⊢ ∃ i, ∀ j ≥ i, padicNorm p (limSeq f j - limSeq f i) < ε
state_after: p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
⊢ ∃ i, ∀ j ≥ i, padicNorm p (limSeq f j - limSeq f i) < ε
tactic: let ⟨N, hN⟩ := exi_rat_seq_conv f hε3
state_before: p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
⊢ ∃ i, ∀ j ≥ i, padicNorm p (limSeq f j - limSeq f i) < ε
state_after: p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
⊢ ∃ i, ∀ j ≥ i, padicNorm p (limSeq f j - limSeq f i) < ε
tactic: let ⟨N2, hN2⟩ := f.cauchy₂ hε3
state_before: p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
⊢ ∃ i, ∀ j ≥ i, padicNorm p (limSeq f j - limSeq f i) < ε
state_after: p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
⊢ ∀ j ≥ max N N2, padicNorm p (limSeq f j - limSeq f (max N N2)) < ε
tactic: exists max N N2
state_before: p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
⊢ ∀ j ≥ max N N2, padicNorm p (limSeq f j - limSeq f (max N N2)) < ε
state_after: p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
j : ℕ
hj : j ≥ max N N2
⊢ padicNorm p (limSeq f j - limSeq f (max N N2)) < ε
tactic: intro j hj
state_before: p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
j : ℕ
hj : j ≥ max N N2
⊢ padicNorm p (limSeq f j - limSeq f (max N N2)) < ε
state_after: p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
j : ℕ
hj : j ≥ max N N2
⊢ padicNormE (↑(limSeq f j) - ↑f (max N N2) + (↑f (max N N2) - ↑(limSeq f (max N N2)))) < ε
tactic: suffices
    padicNormE (limSeq f j - f (max N N2) + (f (max N N2) - limSeq f (max N N2)) : ℚ_[p]) < ε by
    ring_nf at this ⊢
    rw [← padicNormE.eq_padic_norm']
    exact mod_cast this
state_before: p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
j : ℕ
hj : j ≥ max N N2
⊢ padicNormE (↑(limSeq f j) - ↑f (max N N2) + (↑f (max N N2) - ↑(limSeq f (max N N2)))) < ε
state_after: case a
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
j : ℕ
hj : j ≥ max N N2
⊢ padicNormE (↑(limSeq f j) - ↑f (max N N2) + (↑f (max N N2) - ↑(limSeq f (max N N2)))) ≤ ?b

case a
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
j : ℕ
hj : j ≥ max N N2
⊢ ?b < ε

case b
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
j : ℕ
hj : j ≥ max N N2
⊢ ℚ
tactic: apply lt_of_le_of_lt