text: For example, if , the non-terminating decimal representation  and the monotonicity of the rational powers can be used to obtain intervals bounded by rational powers that are as small as desired, and must contain
  So, the upper bounds and the lower bounds of the intervals form two sequences that have the same limit, denoted
  This defines  for every positive  and real  as a continuous function of  and . See also Well-defined expression.
  Exponential function
  The exponential function is often defined as  where  is Euler's number. To avoid circular reasoning, this definition cannot be used here. So, a definition of the exponential function, denoted  and of Euler's number are given, which rely only on exponentiation with positive integer exponents. Then a proof is sketched that, if one uses the definition of exponentiation given in preceding sections, one has
  There are many equivalent ways to define the exponential function, one of them being
  One has  and the exponential identity  holds as well, since
  and the second-order term  does not affect the limit, yielding .
  Euler's number can be defined as . It follows from the preceding equations that  when  is an integer (this results from the repeated-multiplication definition of the exponentiation). If  is real,  results from the definitions given in preceding sections, by using the exponential identity if  is rational, and the continuity of the exponential function otherwise.