File: Mathlib/Topology/Algebra/InfiniteSum/Real.lean

full_name: cauchySeq_of_dist_le_of_summable
code: /-- If the distance between consecutive points of a sequence is estimated by a summable series,
then the original sequence is a Cauchy sequence. -/
theorem cauchySeq_of_dist_le_of_summable (d : ℕ → ℝ) (hf : ∀ n, dist (f n) (f n.succ) ≤ d n)
    (hd : Summable d) : CauchySeq f 
goal: CauchySeq f

full_name: cauchySeq_of_summable_dist
code: theorem cauchySeq_of_summable_dist (h : Summable fun n ↦ dist (f n) (f n.succ)) : CauchySeq f 
goal: CauchySeq f

full_name: dist_le_tsum_of_dist_le_of_tendsto
code: theorem dist_le_tsum_of_dist_le_of_tendsto (d : ℕ → ℝ) (hf : ∀ n, dist (f n) (f n.succ) ≤ d n)
    (hd : Summable d) {a : α} (ha : Tendsto f atTop (𝓝 a)) (n : ℕ) :
    dist (f n) a ≤ ∑' m, d (n + m) 
goal: dist (f n) a ≤ ∑' (m : ℕ), d (n + m)

full_name: dist_le_tsum_of_dist_le_of_tendsto₀
code: theorem dist_le_tsum_of_dist_le_of_tendsto₀ (d : ℕ → ℝ) (hf : ∀ n, dist (f n) (f n.succ) ≤ d n)
    (hd : Summable d) (ha : Tendsto f atTop (𝓝 a)) : dist (f 0) a ≤ tsum d 
goal: dist (f 0) a ≤ tsum d

full_name: dist_le_tsum_dist_of_tendsto
code: theorem dist_le_tsum_dist_of_tendsto (h : Summable fun n ↦ dist (f n) (f n.succ))
    (ha : Tendsto f atTop (𝓝 a)) (n) : dist (f n) a ≤ ∑' m, dist (f (n + m)) (f (n + m).succ) 
goal: dist (f n) a ≤ ∑' (m : ℕ), dist (f (n + m)) (f (n + m).succ)

full_name: dist_le_tsum_dist_of_tendsto₀
code: theorem dist_le_tsum_dist_of_tendsto₀ (h : Summable fun n ↦ dist (f n) (f n.succ))
    (ha : Tendsto f atTop (𝓝 a)) : dist (f 0) a ≤ ∑' n, dist (f n) (f n.succ) 
goal: dist (f 0) a ≤ ∑' (n : ℕ), dist (f n) (f n.succ)

full_name: not_summable_iff_tendsto_nat_atTop_of_nonneg
code: theorem not_summable_iff_tendsto_nat_atTop_of_nonneg {f : ℕ → ℝ} (hf : ∀ n, 0 ≤ f n) :
    ¬Summable f ↔ Tendsto (fun n : ℕ => ∑ i ∈ Finset.range n, f i) atTop atTop 
goal: ¬Summable f ↔ Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, f i) Filter.atTop Filter.atTop

full_name: summable_iff_not_tendsto_nat_atTop_of_nonneg
code: theorem summable_iff_not_tendsto_nat_atTop_of_nonneg {f : ℕ → ℝ} (hf : ∀ n, 0 ≤ f n) :
    Summable f ↔ ¬Tendsto (fun n : ℕ => ∑ i ∈ Finset.range n, f i) atTop atTop 
goal: Summable f ↔ ¬Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, f i) Filter.atTop Filter.atTop

full_name: summable_sigma_of_nonneg
code: theorem summable_sigma_of_nonneg {β : α → Type*} {f : (Σ x, β x) → ℝ} (hf : ∀ x, 0 ≤ f x) :
    Summable f ↔ (∀ x, Summable fun y => f ⟨x, y⟩) ∧ Summable fun x => ∑' y, f ⟨x, y⟩ 
goal: Summable f ↔ (∀ (x : α), Summable fun y => f ⟨x, y⟩) ∧ Summable fun x => ∑' (y : β x), f ⟨x, y⟩

full_name: summable_partition
code: lemma summable_partition {α β : Type*} {f : β → ℝ} (hf : 0 ≤ f) {s : α  → Set β}
    (hs : ∀ i, ∃! j, i ∈ s j) : Summable f ↔
      (∀ j, Summable fun i : s j ↦ f i) ∧ Summable fun j ↦ ∑' i : s j, f i 
goal: Summable f ↔ (∀ (j : α), Summable fun i => f ↑i) ∧ Summable fun j => ∑' (i : ↑(s j)), f ↑i

full_name: summable_prod_of_nonneg
code: theorem summable_prod_of_nonneg {f : (α × β) → ℝ} (hf : 0 ≤ f) :
    Summable f ↔ (∀ x, Summable fun y ↦ f (x, y)) ∧ Summable fun x ↦ ∑' y, f (x, y) 
goal: Summable f ↔ (∀ (x : α), Summable fun y => f (x, y)) ∧ Summable fun x => ∑' (y : β), f (x, y)

full_name: summable_of_sum_le
code: theorem summable_of_sum_le {ι : Type*} {f : ι → ℝ} {c : ℝ} (hf : 0 ≤ f)
    (h : ∀ u : Finset ι, ∑ x ∈ u, f x ≤ c) : Summable f 
goal: Summable f

full_name: summable_of_sum_range_le
code: theorem summable_of_sum_range_le {f : ℕ → ℝ} {c : ℝ} (hf : ∀ n, 0 ≤ f n)
    (h : ∀ n, ∑ i ∈ Finset.range n, f i ≤ c) : Summable f 
goal: Summable f

full_name: Real.tsum_le_of_sum_range_le
code: theorem Real.tsum_le_of_sum_range_le {f : ℕ → ℝ} {c : ℝ} (hf : ∀ n, 0 ≤ f n)
    (h : ∀ n, ∑ i ∈ Finset.range n, f i ≤ c) : ∑' n, f n ≤ c 
goal: ∑' (n : ℕ), f n ≤ c

full_name: tsum_lt_tsum_of_nonneg
code: /-- If a sequence `f` with non-negative terms is dominated by a sequence `g` with summable
series and at least one term of `f` is strictly smaller than the corresponding term in `g`,
then the series of `f` is strictly smaller than the series of `g`. -/
theorem tsum_lt_tsum_of_nonneg {i : ℕ} {f g : ℕ → ℝ} (h0 : ∀ b : ℕ, 0 ≤ f b)
    (h : ∀ b : ℕ, f b ≤ g b) (hi : f i < g i) (hg : Summable g) : ∑' n, f n < ∑' n, g n 
goal: ∑' (n : ℕ), f n < ∑' (n : ℕ), g n