Let 0 < y1 < x1 and set
xn+1 = (1/2)(xn + yn) and yn+1 = sqrt(xnyn)
Prove that 0 < xn+1 - yn+1 < (x1 - y1)/2n for n in N
xn+1 - yn+1 = (1/2)(xn + yn) - sqrt(xnyn) < (1/2)(xn + yn) - y = (1/2)(xn - yn)
Hence by induction and by the fact that 0 < yn < xn for n in N, 0 < xn+1 - yn+1 < (x1 - y1)/2n
What I do not understand is the last part of this explanation. I understand that, by induction, xn+1 - yn+1 < (x1 - y1)/2, but why 2n?
Thanks in advance.