information:extraction
Non-example: rational numbers

The rational numbers Q {\displaystyle \mathbb {Q} } are not complete (for the usual distance): There are sequences of rationals that converge (in R {\displaystyle \mathbb {R} }) to irrational numbers; these are Cauchy sequences having no limit in Q. {\displaystyle \mathbb {Q}.} In fact, if a real number x is irrational, then the sequence (x), 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 R, {\displaystyle \mathbb {R},} for example: