 This proof of the completeness of the real numbers implicitly makes use of the least upper bound axiom.The alternative approach, mentioned above, of the real numbers as the 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).