 XN, respectively. Assume next that the relations in Eq. (R2) apply to any order n > 1. We must then show that the relations also apply to the order n + 1. Then given the nth convergent Xn, i.e., [ao, a1,..., an-1, an], the (n + convergent Xn+1 is of the form [ao, a1,..., an-1, an + 1/an+1]. Applying Eq. (R2), to the latter, we obtain Multiplying the numerator and denominator in Eq. (R6) by an+1 and reordering terms yields qn-2) + qn-1 = Xn+1. The result in Eq. (R7) shows that Xn+1 = Xn+1, corresponding to the suite [ao,al,..., an+1], which proves that the relations in Eq. (R2) apply at order n + 1. If the suite [a0,a],...,AN] is made of positive integers, then Pn, qn are also positive integers and Xn = Pn/qn is a rational number for O<n<N. Property 2 The real numbers Pn, qn defined in Eq. (R2) are co-prime, and satisfy the relation (n > 1): qnPn-1 - Pnqn-1 = (-1)". It is left as an easy exercise first to prove the above by induction. Next, we must show that Pn, qn are co-prime, meaning that their greatest common divisor (GCD) is Assume that k > 1 is some common divisor of Pn, qn, i.e., Pn = kpn and qn = kn, with Pn/qn = Pn/n. Likewise, assume that k' > 1 is some common divisor of Pn- - 1, qn-1, i.e., Pn-1 = k'pn-1 and qn-1 = k'qn-1, with Pn-1/9n-1 = Pn-1/9n-1. Sub- stituting these definitions into Eq. (R8) yields: kk (npn-1 - Pnqn-1) = (-1)". Appendix R Clearly, Pn/qn has the same continued fraction expansion as Pn/qn at all orders n > 1, meaning that from Eq. (R8) we also have nPn-1 - Pnqn-1 = (-1)" and, hence, from Eq. (R9): kk'=1. (R10) Consider next the case n = 1. Since 90 = 1, we have k' = 1 (hence Po, 90 are co-prime) and from Eq. (R10), we have k = 1 (hence p1, q1 are co-prime). The same conclusion applies to the case n = 2 (p2, 92 are co-prime, since p1, q1 are co-prime), and SO on, which proves Property 2, according to which Pn, qn are co-prime at all orders n > 1. Property 3 Given a rational number X, if two integers p, q are such that (R11) then p/q is a convergent of X. We can prove this property as