Math 2.0

Sequences, series, summability Sequences and limits Theorem

Cauchy criterion for sequences

Builds on Cauchy sequenceConvergent sequenceCauchy sequences are boundedBolzano–Weierstrass theorem

Statement

Let (an)(a_n) be a real sequence. Then (an)(a_n) converges if and only if (an)(a_n) is a Cauchy sequence.

Remarks

One direction is convergent sequences are Cauchy. For the other direction, a Cauchy sequence is bounded, so it has a convergent subsequence by the Bolzano–Weierstrass theorem. A Cauchy sequence with a convergent subsequence converges to the same limit. The criterion says that ℝ\mathbb{R} is a complete metric space. It fails in ℚ\mathbb{Q}. Its value is that it proves convergence without a candidate for the limit.

Reading path

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

Show the path as a graph
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 …statementBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementCauchy criterion for sequences, statement, no evidence levelCauchy criterion for …statement
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.
  • no evidence level
  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. Bolzano–Weierstrass theorem statement
  13. Cauchy criterion for sequences statement

Where this leads

52 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cauchy_sequence, Completeness
  2. Web https://en.wikipedia.org/wiki/Limit_of_a_sequence, Cauchy sequences
About this page
Tag0ZA4AX4MW3
URIm2://0ZA4AX4MW3
Kindstatement, theorem
WritingW0 stub
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 0ZA4AX4MW3.
UsesCauchy sequence, Convergent sequence, Cauchy sequences are bounded, Bolzano–Weierstrass theorem

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