Main Index Mathematical Analysis Infinite series and products Sequences
  Subject Index
comment on the page

Cauchy sequence

Given a metric space , a sequence

(1)

is called Cauchy (or fundamental) if for every positive real number there is a positive integer such that for all natural numbers we have .

Consequently, the sequence (1) of complex numbers is Cauchy if for every positive real number there is a positive integer such that for all natural numbers we have , where stands for the absolute value.

Basic properties of Cauchy sequences

• Every convergent sequence is a Cauchy sequence,
• Every Cauchy sequence of real (or complex) numbers is bounded ,
• If in a metric space, a Cauchy sequence possessing a convergent subsequence with limit is itself convergent and has the same limit.

Every convergent sequence is a Cauchy sequence. The converse statement is not true in general. However, in the metric space of complex or real numbers the converse is true.

Theorem: If (1) is a Cauchy sequence of complex or real numbers, then there is a complex or real number , respectively, such that .

In contrast to the above theorem, the property which defines a Cauchy sequence has the advantage that it appears to be merely its “internal” property without an appeal to an “external” object - the limit.   

A metric space in which every Cauchy sequence has a limit in   is called complete.

Cite this web-page as: