Math 2.0

Sequences, series, summability Absolute and conditional convergence, rearrangements, products of series Theorem

Dini's theorem

Builds on Pointwise convergenceUniform convergenceBolzano–Weierstrass theorem

Statement

Let a<ba < b, and let fn ⁣:[a,b]→ℝf_n \colon [a, b] \to \mathbb{R} be continuous functions with fn(x)≤fn+1(x)f_n(x) \le f_{n+1}(x) for all nn and all xx. Assume that (fn)(f_n) converges pointwise to a continuous function ff on [a,b][a, b]. Then (fn)(f_n) converges uniformly to ff on [a,b][a, b].

Remarks

The same holds for decreasing sequences, and for any compact space in place of [a,b][a, b]. One proof uses the Bolzano–Weierstrass theorem. Another uses the open sets {x:f(x)−fn(x)<ε}\{x : f(x) - f_n(x) < \varepsilon\}, which increase and cover [a,b][a, b]. All hypotheses are needed. The sequence xnx^n on [0,1][0, 1] is monotone, but its limit is not continuous. Mathlib states a version as Monotone.tendstoLocallyUniformly_of_forall_tendsto.

History

The theorem is named after Ulisse Dini. He gave the first version in an 1878 book on real functions, printed in Pisa.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementPointwise convergence, definition, no evidence levelPointwise convergencedefinitionEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementUniform convergence, definition, no evidence levelUniform convergencedefinitionDini's theorem, statement, no evidence levelDini's theoremstatement
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. Limit of a sequence definition
  6. Monotone sequence definition
  7. Convergent sequence definition
  8. Monotone convergence theorem for sequences statement
  9. Pointwise convergence definition
  10. Every sequence has a monotone subsequence statement
  11. Bolzano–Weierstrass theorem statement
  12. Uniform convergence definition
  13. Dini's theorem statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Dini%27s_theorem, Formal statement
About this page
Tag0XRRYF1HPE
URIm2://0XRRYF1HPE
Kindstatement, theorem
WritingW1 stated
MSC40A30
StatusThis copy of the site has no build data. Run m2 status 0XRRYF1HPE.
UsesPointwise convergence, Uniform convergence, Bolzano–Weierstrass theorem
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