Math 2.0

Sequences, series, summability Proposition

Cauchy sequences are bounded

Builds on Cauchy sequenceBounded sequence

Statement

Let (an)(a_n) be a Cauchy sequence of real numbers. Then (an)(a_n) is bounded.

Remarks

Take ε=1\varepsilon = 1 to get NN with ∣an−aN∣<1|a_n - a_N| < 1 for all n≥Nn \ge N. Then ∣an∣≤max⁡(∣a0∣,…,∣aN−1∣,∣aN∣+1)|a_n| \le \max(|a_0|, \dots, |a_{N-1}|, |a_N| + 1) for all nn. The statement holds in each metric space. The proof of the Cauchy criterion uses this fact.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionCauchy sequence, definition, no evidence levelCauchy sequencedefinitionCauchy sequences are bounded, statement, no evidence levelCauchy sequences are …statement
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  1. Sequence definition
  2. Real sequence definition
  3. Bounded sequence definition
  4. Cauchy sequence definition
  5. Cauchy sequences are bounded statement

Where this leads

53 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cauchy_sequence, Other properties
About this page
Tag0G6D8E7CTA
URIm2://0G6D8E7CTA
Kindstatement, proposition
WritingW0 stub
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 0G6D8E7CTA.
UsesCauchy sequence, Bounded sequence

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