Theorem: cauchySeq_of_edist_le_of_summable
File: Mathlib/Topology/Instances/ENNReal.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ inst✝ : PseudoEMetricSpace α
f : ℕ → α
d : ℕ → ℝ≥0
hf : ∀ (n : ℕ), edist (f n) (f n.succ) ≤ ↑(d n)
hd : Summable d
⊢ CauchySeq f
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ inst✝ : PseudoEMetricSpace α
f : ℕ → α
d : ℕ → ℝ≥0
hf : ∀ (n : ℕ), edist (f n) (f n.succ) ≤ ↑(d n)
hd : Summable d
ε : ℝ≥0
εpos : 0 < ε
⊢ ∃ N, ∀ (n : ℕ), N ≤ n → edist (f n) (f N) < ↑ε
tactic: refine EMetric.cauchySeq_iff_NNReal.2 fun ε εpos ↦ ?_
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ inst✝ : PseudoEMetricSpace α
f : ℕ → α
d : ℕ → ℝ≥0
hf : ∀ (n : ℕ), edist (f n) (f n.succ) ≤ ↑(d n)
hd : Summable d
ε : ℝ≥0
εpos : 0 < ε
⊢ ∃ N, ∀ (n : ℕ), N ≤ n → edist (f n) (f N) < ↑ε
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ inst✝ : PseudoEMetricSpace α
f : ℕ → α
d : ℕ → ℝ≥0
hf : ∀ (n : ℕ), edist (f n) (f n.succ) ≤ ↑(d n)
ε : ℝ≥0
εpos : 0 < ε
hd : CauchySeq fun n => ∑ x ∈ Finset.range n, d x
⊢ ∃ N, ∀ (n : ℕ), N ≤ n → edist (f n) (f N) < ↑ε
tactic: replace hd : CauchySeq fun n : ℕ ↦ ∑ x ∈ Finset.range n, d x :=
    let ⟨_, H⟩ := hd
    H.tendsto_sum_nat.cauchySeq
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ inst✝ : PseudoEMetricSpace α
f : ℕ → α
d : ℕ → ℝ≥0
hf : ∀ (n : ℕ), edist (f n) (f n.succ) ≤ ↑(d n)
ε : ℝ≥0
εpos : 0 < ε
hd : CauchySeq fun n => ∑ x ∈ Finset.range n, d x
⊢ ∃ N, ∀ (n : ℕ), N ≤ n → edist (f n) (f N) < ↑ε
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ inst✝ : PseudoEMetricSpace α
f : ℕ → α
d : ℕ → ℝ≥0
hf : ∀ (n : ℕ), edist (f n) (f n.succ) ≤ ↑(d n)
ε : ℝ≥0
εpos : 0 < ε
hd : CauchySeq fun n => ∑ x ∈ Finset.range n, d x
N : ℕ
hN : ∀ n ≥ N, dist (∑ x ∈ Finset.range n, d x) (∑ x ∈ Finset.range N, d x) < ↑ε
n : ℕ
hn : N ≤ n
⊢ edist (f n) (f N) < ↑ε
tactic: refine (Metric.cauchySeq_iff'.1 hd ε (NNReal.coe_pos.2 εpos)).imp fun N hN n hn ↦ ?_
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ inst✝ : PseudoEMetricSpace α
f : ℕ → α
d : ℕ → ℝ≥0
hf : ∀ (n : ℕ), edist (f n) (f n.succ) ≤ ↑(d n)
ε : ℝ≥0
εpos : 0 < ε
hd : CauchySeq fun n => ∑ x ∈ Finset.range n, d x
N : ℕ
hN : ∀ n ≥ N, dist (∑ x ∈ Finset.range n, d x) (∑ x ∈ Finset.range N, d x) < ↑ε
n : ℕ
hn : N ≤ n
⊢ edist (f n) (f N) < ↑ε
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ inst✝ : PseudoEMetricSpace α
f : ℕ → α
d : ℕ → ℝ≥0
hf : ∀ (n : ℕ), edist (f n) (f n.succ) ≤ ↑(d n)
ε : ℝ≥0
εpos : 0 < ε
hd : CauchySeq fun n => ∑ x ∈ Finset.range n, d x
N n : ℕ
hn : N ≤ n
hN : dist (∑ x ∈ Finset.range n, d x) (∑ x ∈ Finset.range N, d x) < ↑ε
⊢ edist (f n) (f N) < ↑ε
tactic: specialize hN n hn
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ inst✝ : PseudoEMetricSpace α
f : ℕ → α
d : ℕ → ℝ≥0
hf : ∀ (n : ℕ), edist (f n) (f n.succ) ≤ ↑(d n)
ε : ℝ≥0
εpos : 0 < ε
hd : CauchySeq fun n => ∑ x ∈ Finset.range n, d x
N n : ℕ
hn : N ≤ n
hN : dist (∑ x ∈ Finset.range n, d x) (∑ x ∈ Finset.range N, d x) < ↑ε
⊢ edist (f n) (f N) < ↑ε
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ inst✝ : PseudoEMetricSpace α
f : ℕ → α
d : ℕ → ℝ≥0
hf : ∀ (n : ℕ), edist (f n) (f n.succ) ≤ ↑(d n)
ε : ℝ≥0
εpos : 0 < ε
hd : CauchySeq fun n => ∑ x ∈ Finset.range n, d x
N n : ℕ
hn : N ≤ n
hN :
  ∑ k ∈ Finset.Ico N n, d k < ε ∧
    ∑ x ∈ Finset.range N, d x - (∑ k ∈ Finset.range N, d k + ∑ k ∈ Finset.Ico N n, d k) < ε
⊢ edist (f n) (f N) < ↑ε
tactic: rw [dist_nndist, NNReal.nndist_eq, ← Finset.sum_range_add_sum_Ico _ hn, add_tsub_cancel_left,
    NNReal.coe_lt_coe, max_lt_iff] at hN
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ inst✝ : PseudoEMetricSpace α
f : ℕ → α
d : ℕ → ℝ≥0
hf : ∀ (n : ℕ), edist (f n) (f n.succ) ≤ ↑(d n)
ε : ℝ≥0
εpos : 0 < ε
hd : CauchySeq fun n => ∑ x ∈ Finset.range n, d x
N n : ℕ
hn : N ≤ n
hN :
  ∑ k ∈ Finset.Ico N n, d k < ε ∧
    ∑ x ∈ Finset.range N, d x - (∑ k ∈ Finset.range N, d k + ∑ k ∈ Finset.Ico N n, d k) < ε
⊢ edist (f n) (f N) < ↑ε
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ inst✝ : PseudoEMetricSpace α
f : ℕ → α
d : ℕ → ℝ≥0
hf : ∀ (n : ℕ), edist (f n) (f n.succ) ≤ ↑(d n)
ε : ℝ≥0
εpos : 0 < ε
hd : CauchySeq fun n => ∑ x ∈ Finset.range n, d x
N n : ℕ
hn : N ≤ n
hN :
  ∑ k ∈ Finset.Ico N n, d k < ε ∧
    ∑ x ∈ Finset.range N, d x - (∑ k ∈ Finset.range N, d k + ∑ k ∈ Finset.Ico N n, d k) < ε
⊢ edist (f N) (f n) < ↑ε
tactic: rw [edist_comm]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ inst✝ : PseudoEMetricSpace α
f : ℕ → α
d : ℕ → ℝ≥0
hf : ∀ (n : ℕ), edist (f n) (f n.succ) ≤ ↑(d n)
ε : ℝ≥0
εpos : 0 < ε
hd : CauchySeq fun n => ∑ x ∈ Finset.range n, d x
N n : ℕ
hn : N ≤ n
hN :
  ∑ k ∈ Finset.Ico N n, d k < ε ∧
    ∑ x ∈ Finset.range N, d x - (∑ k ∈ Finset.range N, d k + ∑ k ∈ Finset.Ico N n, d k) < ε
⊢ edist (f N) (f n) < ↑ε
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ inst✝ : PseudoEMetricSpace α
f : ℕ → α
d : ℕ → ℝ≥0
hf : ∀ (n : ℕ), edist (f n) (f n.succ) ≤ ↑(d n)
ε : ℝ≥0
εpos : 0 < ε
hd : CauchySeq fun n => ∑ x ∈ Finset.range n, d x
N n : ℕ
hn : N ≤ n
hN :
  ∑ k ∈ Finset.Ico N n, d k < ε ∧
    ∑ x ∈ Finset.range N, d x - (∑ k ∈ Finset.range N, d k + ∑ k ∈ Finset.Ico N n, d k) < ε
⊢ ∑ i ∈ Finset.Ico N n, ↑(d i) < ↑ε
tactic: refine lt_of_le_of_lt (edist_le_Ico_sum_of_edist_le hn fun _ _ ↦ hf _) ?_
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ inst✝ : PseudoEMetricSpace α
f : ℕ → α
d : ℕ → ℝ≥0
hf : ∀ (n : ℕ), edist (f n) (f n.succ) ≤ ↑(d n)
ε : ℝ≥0
εpos : 0 < ε
hd : CauchySeq fun n => ∑ x ∈ Finset.range n, d x
N n : ℕ
hn : N ≤ n
hN :
  ∑ k ∈ Finset.Ico N n, d k < ε ∧
    ∑ x ∈ Finset.range N, d x - (∑ k ∈ Finset.range N, d k + ∑ k ∈ Finset.Ico N n, d k) < ε
⊢ ∑ i ∈ Finset.Ico N n, ↑(d i) < ↑ε
state_after: no goals
tactic: exact mod_cast hN.1