Editing Cauchy sequence (section)

==Completeness==
A metric space (''X'', ''d'') in which every Cauchy sequence converges to an element of ''X'' is called [[Complete metric space|complete]].

===Examples===
The [[real number]]s are complete under the metric induced by the usual absolute value, and one of the standard [[Construction of the real numbers|constructions of the real numbers]] involves Cauchy sequences of [[rational number]]s. In this construction, each equivalence class of Cauchy sequences of rational numbers with a certain tail behavior—that is, each class of sequences that get arbitrarily close to one another— is a real number.

A rather different type of example is afforded by a metric space ''X'' which has the [[discrete space|discrete metric]] (where any two distinct  points are at distance 1 from each other). Any Cauchy sequence of elements of ''X'' must be constant beyond some fixed point, and converges to the eventually repeating term.

===Non-example: rational numbers===
The [[rational number]]s <math>\Q</math> are not complete (for the usual distance):<br/>
There are sequences of rationals that converge (in <math>\R</math>) to [[irrational number]]s; these are Cauchy sequences having no limit in <math>\Q.</math> In fact, if a real number ''x'' is irrational, then the sequence (''x''<sub>''n''</sub>), whose ''n''-th term is the truncation to ''n'' decimal places of the decimal expansion of ''x'', gives a Cauchy sequence of rational numbers with irrational limit ''x''. Irrational numbers certainly exist in <math>\R,</math> for example:

* The sequence defined by <math>x_0=1, x_{n+1}=\frac{x_n+2/x_n}{2}</math> consists of rational numbers (1, 3/2, 17/12,...), which is clear from the definition; however it converges to the irrational [[square root of 2]], see [[Methods of computing square roots#Heron's method|Babylonian method of computing square root]].
* The sequence <math>x_n = F_n / F_{n-1}</math> of ratios of consecutive [[Fibonacci number]]s which, if it converges at all, converges to a limit <math>\phi</math> satisfying <math>\phi^2 = \phi+1,</math> and no rational number has this property.  If one considers this as a sequence of real numbers, however, it converges to the real number <math>\varphi = (1+\sqrt5)/2,</math> the [[Golden ratio]], which is irrational.
* The values of the exponential, sine and cosine functions, exp(''x''), sin(''x''), cos(''x''), are known to be irrational for any rational value of <math>x \neq 0,</math> but each can be defined as the limit of a rational Cauchy sequence, using, for instance, the [[Maclaurin series]].

===Non-example: open interval===
The open interval <math>X = (0, 2)</math> in the set of real numbers with an ordinary distance in <math>\R</math> is not a complete space: there is a sequence <math>x_n = 1/n</math> in it, which is Cauchy (for arbitrarily small distance bound <math>d > 0</math> all terms <math>x_n</math> of <math>n > 1/d</math> fit in the <math>(0, d)</math> interval), however does not converge in <math>X</math> — its 'limit', number 0, does not belong to the space <math>X .</math>

===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.