Word.ws English Dictionary

Cauchy sequence

Cauchy sequence noun: any sequence x_n in a metric space with metric d such that for every ϵ>0 there exists a natural number N such that for all k,m>N, d(x_k,x_m)<ϵ.

Noun

Cauchy sequence (plural Cauchy sequences)

  1. Any sequence x_n in a metric space with metric d such that for every ϵ>0 there exists a natural number N such that for all k,m>N, d(x_k,x_m)<ϵ.
    • However, it is possible to derive topological results from statements about Cauchy sequences; for example, a subset A of the space of real numbers is closed if and only if each Cauchy sequence in A converges to some point of A.1955, [Van Nostrand], John L. Kelley, General Topology, Springer, published 1975, page 174
    • In the case of the real line, every Cauchy sequence converges; that is, being a Cauchy sequence is sufficient to guarantee the existence of a limit. In the general case, however, this is not so. If a metric space does have the property that every Cauchy sequence converges, the space is called a...2000, George Bachman, Lawrence Narici, Functional Analysis, page 52

Cauchy sequence meaning in Tamil

Cauchy sequence in Tamil

கௌசி வரிசை

Examples of Cauchy sequence in a sentence

  • In the case of the real line, every Cauchy sequence converges; that is, being a Cauchy sequence is sufficient to guarantee the existence of a limit.2000, George Bachman, Lawrence Narici, Functional Analysis, page 52:
  • Cantor first redefined Cauchy sequences using rational numbers only.[…]Cantor's idea was to define the real number line as the collection of all (rational) Cauchy sequences.2012, David Applebaum, Limits, Limits Everywhere: The Tools of Mathematical Analysis, Oxford University Press, page 153:

Forms of Cauchy sequence

pluralCauchy sequences

Origin of Cauchy sequence

Named after French mathematician Augustin-Louis Cauchy (1789–1857), who made pioneering contributions to analysis.

Words formed from Cauchy sequence

Questions about Cauchy sequence

What does Cauchy sequence mean?
Cauchy sequence means any sequence x_n in a metric space with metric d such that for every ϵ>0 there exists a natural number N such that for all k,m>N, d(x_k,x_m)<ϵ.
What is Cauchy sequence in Tamil?
Cauchy sequence in Tamil is கௌசி வரிசை. See Cauchy sequence meaning in Tamil.
How do you use Cauchy sequence in a sentence?
In the case of the real line, every Cauchy sequence converges; that is, being a Cauchy sequence is sufficient to guarantee the existence of a limit.
What part of speech is Cauchy sequence?
Cauchy sequence is a noun.
What is the plural of Cauchy sequence?
The plural of Cauchy sequence is Cauchy sequences.

Words near Cauchy sequence in the dictionary

  1. Cauchy distributions
  2. Cauchy horizon
  3. Cauchy horizons
  4. Cauchy problem
  5. Cauchy problems
  6. Cauchy sequence
  7. Cauchy sequences
  8. Cauchy space
  9. Cauchy spaces
  10. Cauchy-Riemann equation
  11. Cauchy-Riemann equations

Source and license

Adapted from Wiktionary: Cauchy sequence (page history and authors); cauchy sequence (Tamil Wiktionary) (page history and authors). Text available under the Creative Commons Attribution-ShareAlike 4.0 license. Word.ws reformatted the entry and added links. Wiktionary data of 2026-09-02.

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