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Cauchy sequence

Given a metric space typeset structure, a sequence

x _ 1, x _ 2, x _ 3, ...(1)

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

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

Basic properties of Cauchy sequences

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 typeset structure, respectively, such that typeset structure.

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 typeset structure in which every Cauchy sequence has a limit in typeset structure  is called complete.

Cite this web-page as:

Štefan Porubský: Cauchy sequence.

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