Theorems · Theorem · sequences and series
dist_le_of_le_geometric_two_of_tendsto
∀ {α : Type u_1} [inst : PseudoMetricSpace α] {C : ℝ} {f : ℕ → α},
(∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C / 2 / 2 ^ n) →
∀ {a : α}, Filter.Tendsto f Filter.atTop (nhds a) → ∀ (n : ℕ), dist (f n) a ≤ C / 2 ^ nIf dist (f n) (f (n+1)) is bounded by (C / 2) / 2^n, then the distance from
f n to the limit of f is bounded above by C / 2^n.
- Defined in
- Mathlib.Analysis.SpecificLimits.Basic
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 156 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- PseudoMetricSpace
Around this declaration
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Cites15
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
- nhdsstatement and proof · cited by 5,554
- Filter.Tendstostatement and proof · cited by 3,814
- Filter.atTopstatement and proof · cited by 2,405
- SummationFilter.unconditionalproof · cited by 2,068
- PseudoMetricSpacestatement and proof · cited by 1,550
- Dist.diststatement and proof · cited by 1,539
- add_commproof · cited by 1,535
- tsumproof · cited by 1,148
- pow_addproof · cited by 315
- HasSum.tsum_eqproof · cited by 150
- HasSum.div_constproof · cited by 18
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