Theorems · Theorem · sequences and series
tendsto_pow_atTop_nhds_zero_of_norm_lt_one
∀ {R : Type u_2} [inst : SeminormedRing R] {x : R}, ‖x‖ < 1 → Filter.Tendsto (fun n => x ^ n) Filter.atTop (nhds 0)In a normed ring, the powers of an element x with ‖x‖ < 1 tend to zero.
- Defined in
- Mathlib.Analysis.SpecificLimits.Normed
- Cited by
- 13 results in Mathlib
- Foundations
- Depth 122 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SeminormedRing
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement · cited by 25,697
- nhdsstatement · cited by 5,554
- Norm.normstatement and proof · cited by 5,413
- Filter.Tendstostatement · cited by 3,814
- Filter.atTopstatement · cited by 2,405
- norm_nonnegproof · cited by 725
- SeminormedRingstatement and proof · cited by 446
- tendsto_pow_atTop_nhds_zero_of_lt_oneproof · cited by 18
- eventually_norm_pow_leproof · cited by 4
- squeeze_zero_norm'proof · cited by 4
Cited by13
Results whose statement or proof uses this declaration.
- hasSum_geometric_of_norm_lt_oneproof · cited by 4
- tendsto_pow_atTop_nhds_zero_iff_norm_lt_oneproof · cited by 3
- tendsto_pow_atTop_nhds_zero_of_abs_lt_oneproof · cited by 3
- FiniteDimensional.of_totallyBounded_nhds_zeroproof · cited by 3
- ContinuousSMul.topology_eq_of_nhds_inf_principal_eqproof · cited by 1
- summable_norm_pow_mul_geometric_div_one_subproof · cited by 1
- continuum_le_cardinal_of_nontriviallyNormedFieldproof · cited by 1
- LinearMap.continuousAt_zero_of_locally_boundedproof · cited by 1
- Complex.abel_auxproof · cited by 1
- cardinal_eq_of_mem_nhds_zeroproof · cited by 1
- tendsto_zero_geometric_tsum_pnatproof · cited by 0
- mem_tangentConeAt_of_pow_smulproof · cited by 0