Cauchy sequence

Etymology

Definitions

1. 1

   Any sequence x_n in a metric space with metric d such that for every ϵ>0 there exists a…

   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.
   • 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.

The neighborhood