Theorems · Theorem · general topology

edist_le_Ico_sum_of_edist_le

∀ {α : Type u} [inst : PseudoEMetricSpace α] {f : ℕ → α} {m n : ℕ},
  m ≤ n →
    ∀ {d : ℕ → ENNReal},
      (∀ {k : ℕ}, m ≤ k → k < n → edist (f k) (f (k + 1)) ≤ d k) → edist (f m) (f n) ≤ ∑ i ∈ Finset.Ico m n, d i

A version of edist_le_Ico_sum_edist with each intermediate distance replaced with an upper estimate.

Defined in: Mathlib.Topology.EMetricSpace.Basic
Cited by: 3 results in Mathlib
Foundations: Depth 148 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: PseudoEMetricSpace

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Cites9

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

ENNRealstatement and proof · cited by 9,879
Finset.sumstatement · cited by 5,195
PseudoEMetricSpacestatement and proof · cited by 1,536
le_transproof · cited by 985
EDist.ediststatement and proof · cited by 735
Finset.Icostatement and proof · cited by 450
Finset.sum_le_sumproof · cited by 182
Finset.mem_Icoproof · cited by 50
edist_le_Ico_sum_edistproof · cited by 2

Cited by3

Results whose statement or proof uses this declaration.

cauchySeq_of_edist_le_of_summableproof · cited by 2
edist_le_tsum_of_edist_le_of_tendstoproof · cited by 2
edist_le_range_sum_of_edist_leproof · cited by 0