Editing Cauchy sequence (section)

===Other properties===
* Every convergent sequence (with limit ''s'', say) is a Cauchy sequence, since, given any real number <math>\varepsilon > 0,</math> beyond some fixed point, every term of the sequence is within distance <math>\varepsilon/2</math> of ''s'', so any two terms of the sequence are within distance <math>\varepsilon</math> of each other.
* In any metric space, a Cauchy sequence <math>x_n</math> is [[Bounded function|bounded]] (since for some ''N'', all terms of the sequence from the ''N''-th onwards are within distance 1 of each other, and if ''M'' is the largest distance between <math>x_N</math> and any terms up to the ''N''-th, then no term of the sequence has distance greater than <math>M + 1</math> from <math>x_N</math>).
* In any metric space, a Cauchy sequence which has a convergent subsequence with limit ''s'' is itself convergent (with the same limit), since, given any real number ''r'' > 0, beyond some fixed point in the original sequence, every term of the subsequence is within distance ''r''/2 of ''s'', and any two terms of the original sequence are within distance ''r''/2 of each other, so every term of the original sequence is within distance ''r'' of ''s''.

These last two properties, together with the [[Bolzano–Weierstrass theorem]], yield one standard proof of the completeness of the real numbers, closely related to both the Bolzano–Weierstrass theorem and the [[Heine–Borel theorem]]. Every Cauchy sequence of real numbers is bounded, hence by Bolzano–Weierstrass has a convergent subsequence, hence is itself convergent. This proof of the completeness of the real numbers implicitly makes use of the [[least upper bound axiom]]. The alternative approach, mentioned above, of {{em|constructing}} the real numbers as the [[Completion (metric space)|completion]] of the rational numbers, makes the completeness of the real numbers tautological.

One of the standard illustrations of the advantage of being able to work with Cauchy sequences and make use of completeness is provided by consideration of the summation of an [[infinite series]] of real numbers
(or, more generally, of elements of any complete [[normed linear space]], or [[Banach space]]).  Such a series 
<math display="inline">\sum_{n=1}^{\infty} x_n</math> is considered to be convergent if and only if the sequence of [[partial sum]]s <math>(s_{m})</math> is convergent, where  <math display="inline">s_m = \sum_{n=1}^{m} x_n.</math> It is a routine matter to determine whether the sequence of partial sums is Cauchy or not, since for positive integers <math>p > q,</math>
<math display="block">s_p - s_q = \sum_{n=q+1}^p x_n.</math>

If <math>f : M \to N</math> is a [[uniformly continuous]] map between the metric spaces ''M'' and ''N'' and (''x''<sub>''n''</sub>) is a Cauchy sequence in ''M'', then <math>(f(x_n))</math> is a Cauchy sequence in ''N''. If <math>(x_n)</math> and <math>(y_n)</math> are two Cauchy sequences in the rational, real or complex numbers, then the sum <math>(x_n + y_n)</math> and the product <math>(x_n y_n)</math> are also Cauchy sequences.