Thread: Contractive Sequence

Let x1 be greater than zero. x(n+1)=1/(2+xn) for all n greater than or equal to 2. Show that (xn) is a contractive sequence and find its limit.

Stuck on the proof. Pretty sure the limit would just be the roots of (x^2)+2x-1=0. Any help appreciated.

I just need a little help. I have that:
abs(x(n+1)-x(n))=abs(1/(2+xn)-1(2+x(n-1))).

I just don't know how to simplify this in terms of abs(xn-x(n-1)) and find what C in the contractive sequence is.

Could anyone just steer me in the right direction?

We need to show,

For some
Now,
Let then we have,



Thus,

If und only if,

If und only if,

Thus, by choosing this konstant to be that value we can form a contractive sequence.