section_path: Properties/Sequences of real numbers/Cauchy sequences
section_name: Cauchy sequences
paragraph_text: Cauchy sequences
  
  A property of convergent sequences of real numbers is that they are Cauchy sequences. The definition of a Cauchy sequence { a n } {\displaystyle \{a_{n}\}} is that for every real number ϵ > 0 {\displaystyle \epsilon >0}, there is an N {\displaystyle N} such that whenever m, n > N {\displaystyle m,n>N}, | a m − a n | < ϵ. {\displaystyle |a_{m}-a_{n}|<\epsilon.}
  
  Informally, for any arbitrarily small error ϵ {\displaystyle \epsilon }, it is possible to find an interval of diameter ϵ {\displaystyle \epsilon } such that eventually the sequence is contained within the interval.
  
  Cauchy sequences are closely related to convergent sequences. In fact, for sequences of real numbers they are equivalent: any Cauchy sequence is convergent.
  
  In general metric spaces, it continues to hold that convergent sequences are also Cauchy. But the converse is not true: not every Cauchy sequence is convergent in a general metric space. A classic counterexample is the rational numbers, Q {\displaystyle \mathbb {Q} }, with the usual distance. The sequence of decimal approximations to 2 {\displaystyle {\sqrt {2}}}, truncated at the n {\displaystyle n} th decimal place is a Cauchy sequence, but does not converge in Q {\displaystyle \mathbb {Q} }.
  
  A metric space in which every Cauchy sequence is also convergent, that is, Cauchy sequences are equivalent to convergent sequences, is known as a complete metric space.
  
  One reason Cauchy sequences can be "easier to work with" than convergent sequences is that they are a property of the sequence { a n } {\displaystyle \{a_{n}\}} alone, while convergent sequences require not just the sequence { a n } {\displaystyle \{a_{n}\}} but also the limit of the sequence a {\displaystyle a}.