Math 2.0

statement theorem

Cauchy criterion for sequences

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.

Explanations

L1 human

To show that a sequence has a limit, you often need to know the limit first. The Cauchy criterion avoids this. It is enough to show that the terms become close to each other.

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To read this node, read these 13 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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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. Bolzano–Weierstrass theorem statement
  13. Cauchy criterion for sequences statement

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37 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Cauchy_sequence, Completeness
  2. Web https://en.wikipedia.org/wiki/Limit_of_a_sequence, Cauchy sequences

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