
Differentiation Analysis
Hi,
Let f: R > R and differentiable on R. Suppose sup{f'(x)} = M < 1. Take S ele R and define a sequence Sn+1 = f(Sn) n = 0,1,2,3,... Show that {Sn} is a cauchy sequence. Hint show first that Sn+1  Sn<= MSn  Sn1.
So I know you can start with the mean value theorem: f'(x) = (Sn+1  Sn)/(Sn  Sn1). Then (Sn  Sn1)f'(x) = Sn+1  Sn.
From here i'm not exactly sure what to do.
Thanks
For those of you who have the kenneth ross book it is exercize 29.18.

Theorem: If $\displaystyle 0<a<1$ and $\displaystyle s_{n+1}s_n \leq as_n  s_{n1}$. Then the sequence is convergent (or Cauchy).
Proof: Here.