Convergence of an iterative sequence

Consider the recursive relation,

x_k = f(x_{k-1}) + f(x_{k-2})

where f is non-negative, strictly decreasing and concave (f''>0) with
f(0)=1 and f(inf)=0.

Also, x_0=0

I would need to know if this always converges to the fixed point
x=2f(x), and why. How to solve this kind of problems? Any help would
be appreciated.

Thanks in advance,

Jesús Cid.
.