text: It can be seen that  is 1 and  (which exists by ) is a negative integer. Thereafter, the  alternate in sign and strictly increase in magnitude, which follows inductively from the definitions and the fact that  for , the case  holds because . The same is true for the  after the first few terms, for the same reason. Furthermore, it is easy to see that  (when a and b are both positive and ). Thus, noticing that , we obtain
  This, accompanied by the fact that  are larger than or equal to in absolute value than any previous  or  respectively completed the proof.
  Polynomial extended Euclidean algorithm
  For univariate polynomials with coefficients in a field, everything works similarly, Euclidean division, Bézout's identity and extended Euclidean algorithm. The first difference is that, in the Euclidean division and the algorithm, the inequality  has to be replaced by an inequality on the degrees  Otherwise, everything which precedes in this article remains the same, simply by replacing integers by polynomials.
  A second difference lies in the bound on the size of the Bézout coefficients provided by the extended Euclidean algorithm, which is more accurate in the polynomial case, leading to the following theorem.
  If a and b are two nonzero polynomials, then the extended Euclidean algorithm produces the unique pair of polynomials (s, t) such that
  and
  A third difference is that, in the polynomial case, the greatest common divisor is defined only up to the multiplication by a non zero constant. There are several ways to define unambiguously a greatest common divisor.