Math 2.0

General topology Proposition

Epsilon characterisation of Cauchy sequences

Builds on Metric spaceDiameter of a set

Statement

Let (X,d)(X, d) be a metric space and let (xn)n≥1(x_n)_{n \ge 1} be a sequence in XX. For each N≥1N \ge 1, let TN={xn:n≥N}T_N = \{x_n : n \ge N\}. Then the following two conditions are equivalent:

  1. For each real ε>0\varepsilon > 0 there is an integer NN such that d(xm,xn)<εd(x_m, x_n) < \varepsilon for all m,n≥Nm, n \ge N.
  2. The diameters diam⁡(TN)\operatorname{diam}(T_N) tend to 00 as N→∞N \to \infty.

The sequence (xn)(x_n) is a Cauchy sequence when it satisfies these conditions.

Remarks

Condition 1 is the usual definition of a Cauchy sequence. Each convergent sequence is a Cauchy sequence: if xn→xx_n \to x, then d(xm,xn)≤d(xm,x)+d(x,xn)d(x_m, x_n) \le d(x_m, x) + d(x, x_n). In ℚ\mathbb{Q} with the usual metric, the decimal approximations 1,1.4,1.41,1.414,…1, 1.4, 1.41, 1.414, \ldots of 2\sqrt{2} form a Cauchy sequence that does not converge in ℚ\mathbb{Q}. Each Cauchy sequence is bounded.

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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 spacedefinitionDiameter of a set, definition, no evidence levelDiameter of a setdefinitionEpsilon characterisation of Cauchy sequences, statement, no evidence levelEpsilon characterisat…statement
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  2. Extended real number line object
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  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. Diameter of a set definition
  20. Epsilon characterisation of Cauchy sequences statement

Where this leads

87 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cauchy_sequence, In a metric space
About this page
Tag0MCK8K2D7Z
URIm2://0MCK8K2D7Z
Kindstatement, proposition
WritingW0 stub
MSC54E35
StatusThis copy of the site has no build data. Run m2 status 0MCK8K2D7Z.
UsesMetric space, Diameter of a set

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