Theorems · Theorem · measure theory
cauchySeq_of_edist_le_of_summable
∀ {α : Type u_1} [inst : PseudoEMetricSpace α] {f : ℕ → α} (d : ℕ → NNReal),
(∀ (n : ℕ), edist (f n) (f n.succ) ≤ ↑(d n)) → Summable d → CauchySeq fIf the extended distance between consecutive points of a sequence is estimated
by a summable series of NNReals, then the original sequence is a Cauchy sequence.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 149 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- PseudoEMetricSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites29
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realproof · cited by 25,697
- ENNRealstatement and proof · cited by 9,879
- Finset.sumproof · cited by 5,195
- NNRealstatement and proof · cited by 4,310
- SummationFilter.unconditionalstatement and proof · cited by 2,068
- Dist.distproof · cited by 1,539
- PseudoEMetricSpacestatement and proof · cited by 1,536
- Finset.rangeproof · cited by 1,341
- ENNReal.ofNNRealstatement and proof · cited by 1,279
- NNReal.toRealproof · cited by 1,260
- Summablestatement and proof · cited by 778
- EDist.ediststatement and proof · cited by 735
Cited by2
Results whose statement or proof uses this declaration.
- cauchySeq_of_dist_le_of_summableproof · cited by 3
- cauchySeq_of_edist_le_of_tsum_ne_topproof · cited by 1