Meaning of Cauchy sequence | Babel Free

Definitions

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.

Equivalents

Examples

“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.”

“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.”

“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.”

CEFR level