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)].
The Attempt at a Solution
Proof by induction.
For n=1, abs(x(3)-x(2))<= 1/2 abs(x(2)-x(1))
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!