Cauchy sequence
Noun
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Noun
Cauchy sequence (plural Cauchy sequences)
- (analysis) Any sequence x_n in a metric space with metric d such that for every \epsilon > 0 there exists a natural number N such that for all k, m \ge N , d(x_k, x_m) < \epsilon .
- 1955, [Van Nostrand], John L. Kelley, General Topology, 1975, Springer, page 174 ↗,
- 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.
- 2000, George Bachman, Lawrence Narici, Functional Analysis, page 52 ↗,
- 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 complete metric space.
- 2012, David Applebaum, Limits, Limits Everywhere: The Tools of Mathematical Analysis, Oxford University Press, page 153 ↗,
- 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.
- 1955, [Van Nostrand], John L. Kelley, General Topology, 1975, Springer, page 174 ↗,
- Cauchy convergence
- Cauchy filter
- Cauchy net
- Cauchy space
- Italian: successione di Cauchy
This text is extracted from the Wiktionary and it is available under the CC BY-SA 3.0 license | Terms and conditions | Privacy policy 0.006