1. The problem statement, all variables and given/known data Consider the sequence (x(n)) defined recursively by x(1)=1 and x(n+1)=1+1/x(n). Prove that abs[x(n+2)-x(n+1)]<= 1/2 abs[x(n+1)-x(n)]. 2. Relevant equations 3. The attempt at a solution Proof by induction. For n=1, abs(x(3)-x(2))<= 1/2 abs(x(2)-x(1)) abs(-1/2)<=1/2 abs(1) 1/2=1/2 So, for n=1 is true. Suppose that n=k and that abs[x(k+2)-x(k+1)]<= 1/2 abs[x(k+1)-x(k)]. We have to show that abs[x((k+1)+2)-x((k+1)+1)]<= 1/2 abs[x((k+1)+1)-x(k+1)]. I'm not sure if this is correct. If so, how can I continue? Thanks in advance!