Math 2.0

Sequences, series, summability Definition

Uniformly Cauchy sequence of functions

Builds on Cauchy sequenceUniform convergence on a setCauchy criterion for sequences

Definition

Let SS be a set and let fn ⁣:S→ℝf_n \colon S \to \mathbb{R} for n≥1n \ge 1. The sequence (fn)(f_n) is uniformly Cauchy on SS if for every ε>0\varepsilon > 0 there is NN such that ∣fn(x)−fm(x)∣<εfor all m,n≥N and all x∈S.|f_n(x) - f_m(x)| < \varepsilon \quad \text{for all } m, n \ge N \text{ and all } x \in S.

Remarks

For each xx, the sequence (fn(x))(f_n(x)) is then a Cauchy sequence. So it has a limit f(x)f(x), by the Cauchy criterion. A sequence of real functions is uniformly Cauchy on SS if and only if it converges uniformly on SS. This criterion proves uniform convergence without knowing the limit. Mathlib defines it as UniformCauchySeqOn.

Reading path

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionCauchy sequence, definition, no evidence levelCauchy sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionCauchy sequences are bounded, statement, no evidence levelCauchy sequences are …statementMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementPointwise convergence, definition, no evidence levelPointwise convergencedefinitionBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementUniform convergence, definition, no evidence levelUniform convergencedefinitionCauchy criterion for sequences, statement, no evidence levelCauchy criterion for …statementUniform convergence on a set, definition, no evidence levelUniform convergence o…definitionUniformly Cauchy sequence of functions, definition, no evidence levelUniformly Cauchy sequ…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Sequence definition
  2. Real sequence definition
  3. Subsequence definition
  4. Bounded sequence definition
  5. Cauchy sequence definition
  6. Limit of a sequence definition
  7. Monotone sequence definition
  8. Cauchy sequences are bounded statement
  9. Monotone convergence theorem for sequences statement
  10. Convergent sequence definition
  11. Every sequence has a monotone subsequence statement
  12. Pointwise convergence definition
  13. Bolzano–Weierstrass theorem statement
  14. Uniform convergence definition
  15. Cauchy criterion for sequences statement
  16. Uniform convergence on a set definition
  17. Uniformly Cauchy sequence of functions definition

Where this leads

47 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Uniform_convergence, Properties
About this page
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Kinddefinition
WritingW0 stub
MSC40A30
DefinesUniformly Cauchy sequence of functions
UsesCauchy sequence, Uniform convergence on a set, Cauchy criterion for sequences

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