Editing Cauchy sequence (section)

===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]].