section_path: Topological completeness
section_name: Topological completeness
paragraph_text: Topological completeness
  
  A main reason for using real numbers is so that many sequences have limits. More formally, the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section):
  
  A sequence (x) of real numbers is called a Cauchy sequence if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance | x − x | is less than ε for all n and m that are both greater than N. This definition, originally provided by Cauchy, formalizes the fact that the x eventually come and remain arbitrarily close to each other.
  
  A sequence (x) converges to the limit x if its elements eventually come and remain arbitrarily close to x, that is, if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance | x − x | is less than ε for n greater than N.
  
  Every convergent sequence is a Cauchy sequence, and the converse is true for real numbers, and this means that the topological space of the real numbers is complete.
  
  The set of rational numbers is not complete. For example, the sequence (1; 1.4; 1.41; 1.414; 1.4142; 1.41421;...), where each term adds a digit of the decimal expansion of the positive square root of 2, is Cauchy but it does not converge to a rational number (in the real numbers, in contrast, it converges to the positive square root of 2).
  
  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. This proves that the sequence is Cauchy, and thus converges, showing that e x {\displaystyle e^{x}} is well defined for every x.