Math 2.0

General topology Proposition

Completeness via Cauchy sequences

Builds on Complete metric spaceEpsilon characterisation of Cauchy sequencesConvergence of a sequence in a metric space

Statement

Let (X,d)(X, d) be a metric space. Then XX is complete if and only if each Cauchy sequence in XX has a convergent subsequence.

Remarks

The key fact: if a Cauchy sequence (xn)(x_n) has a subsequence that converges to xx, then xn→xx_n \to x. For ε>0\varepsilon > 0, choose NN for ε/2\varepsilon / 2 in the Cauchy condition and a term xnkx_{n_k} with nk≥Nn_k \ge N and d(xnk,x)<ε/2d(x_{n_k}, x) < \varepsilon / 2. A consequence: a metric space in which each sequence has a convergent subsequence is complete. Thus each sequentially compact metric space is complete.

Reading path

To understand this page from the ground up, read these 23 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionExtended real number line, object, no evidence levelExtended real number …objectProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionSupremum, definition, no evidence levelSupremumdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionConvergence of a sequence in a metric space, definition, no evidence levelConvergence of a sequ…definitionDiameter of a set, definition, no evidence levelDiameter of a setdefinitionEpsilon characterisation of Cauchy sequences, statement, no evidence levelEpsilon characterisat…statementComplete metric space, definition, no evidence levelComplete metric spacedefinitionCompleteness via Cauchy sequences, statement, no evidence levelCompleteness via Cauc…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. Element definition
  2. Extended real number line object
  3. Proposition concept
  4. Real numbers object
  5. Set bounded above definition
  6. Set definition
  7. Implication definition
  8. Predicate definition
  9. Supremum definition
  10. Ordered pair definition
  11. Existential quantifier definition
  12. Universal quantifier definition
  13. Cartesian product definition
  14. Subset definition
  15. Uniqueness quantifier definition
  16. Binary relation definition
  17. Function definition
  18. Metric space definition
  19. Convergence of a sequence in a metric space definition
  20. Diameter of a set definition
  21. Epsilon characterisation of Cauchy sequences statement
  22. Complete metric space definition
  23. Completeness via Cauchy sequences statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Cauchy_sequence, In a metric space
  2. Web https://en.wikipedia.org/wiki/Complete_metric_space, Definition
About this page
Tag0N1ZWFJDN5
URIm2://0N1ZWFJDN5
Kindstatement, proposition
WritingW0 stub
MSC54E50
StatusThis copy of the site has no build data. Run m2 status 0N1ZWFJDN5.
UsesComplete metric space, Epsilon characterisation of Cauchy sequences, Convergence of a sequence in a metric space

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