Theorems · Theorem · general topology
cauchySeq_tendsto_of_complete
∀ {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] [inst : Preorder β] [CompleteSpace α] {u : β → α},
CauchySeq u → ∃ x, Filter.Tendsto u Filter.atTop (nhds x)A Cauchy sequence in a complete space converges
- Defined in
- Mathlib.Topology.UniformSpace.Cauchy
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 62 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Preorderstatement and proof · cited by 7,952
- nhdsstatement · cited by 5,554
- Filter.Tendstostatement · cited by 3,814
- CompleteSpacestatement and proof · cited by 2,532
- Filter.atTopstatement · cited by 2,405
- UniformSpacestatement and proof · cited by 2,040
- CauchySeqstatement and proof · cited by 131
- CompleteSpace.completeproof · cited by 19
Cited by10
Results whose statement or proof uses this declaration.
- ContractingWith.exists_fixedPointproof · cited by 4
- ApproximatesLinearOn.surjOn_closedBall_of_nonlinearRightInverseproof · cited by 3
- MeasureTheory.tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioiproof · cited by 2
- MeasureTheory.completeSpace_of_completeSpace_Lpproof · cited by 1
- MeasureTheory.Lp.ae_tendsto_of_cauchy_eLpNormproof · cited by 1
- CantorScheme.ClosureAntitone.map_of_vanishingDiamproof · cited by 1
- Antitone.tendsto_alternating_series_of_tendsto_zeroproof · cited by 1
- NormedAddCommGroup.summable_imp_tendsto_of_completeproof · cited by 1
- controlled_closure_of_completeproof · cited by 1
- Monotone.tendsto_alternating_series_of_tendsto_zeroproof · cited by 0