Theorems · Theorem · general topology
tendsto_iff_dist_tendsto_zero
∀ {α : Type u} {β : Type v} [inst : PseudoMetricSpace α] {f : β → α} {x : Filter β} {a : α},
Filter.Tendsto f x (nhds a) ↔ Filter.Tendsto (fun b => dist (f b) a) x (nhds 0)- Defined in
- Mathlib.Topology.MetricSpace.Pseudo.Defs
- Cited by
- 16 results in Mathlib
- Foundations
- Depth 118 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- PseudoMetricSpace
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.
- Realstatement and proof · cited by 25,697
- Filterstatement and proof · cited by 8,121
- nhdsstatement and proof · cited by 5,554
- Filter.Tendstostatement and proof · cited by 3,814
- PseudoMetricSpacestatement and proof · cited by 1,550
- Dist.diststatement and proof · cited by 1,539
- Filter.tendsto_comap_iffproof · cited by 55
- nhds_comap_distproof · cited by 3
Cited by16
Results whose statement or proof uses this declaration.
- tendsto_iff_norm_sub_tendsto_zeroproof · cited by 17
- MeasureTheory.Lp.tendsto_Lp_iff_tendsto_eLpNorm'proof · cited by 5
- ApproximatesLinearOn.surjOn_closedBall_of_nonlinearRightInverseproof · cited by 3
- continuousAt_of_locally_lipschitzproof · cited by 3
- BoundedContinuousFunction.exists_extension_norm_eq_of_isClosedEmbedding'proof · cited by 2
- MeasureTheory.Lp.simpleFunc.denseRange_coeSimpleFuncNonnegToLpNonnegproof · cited by 1
- aemeasurable_of_unif_approxproof · cited by 1
- continuous_iff_continuous_distproof · cited by 1
- MeasureTheory.tendsto_indicatorConstLp_setproof · cited by 1
- MeasureTheory.measurableSet_range_of_continuous_injectiveproof · cited by 1
- CircleDeg1Lift.tendsto_translation_number₀'proof · cited by 1