 The completeness property of the reals is the basis on which calculus, and, more generally mathematical analysis are built.In particular, the test that a sequence is a Cauchy sequence allows proving that a sequence has a limit, without computing it, and even without knowing it.For example, the standard series of the exponential function

converges to a real number for every "x", because the sums

can be made arbitrarily small (independently of "M") by choosing "N" sufficiently large.