Theorem: If and . Then the sequence is convergent (or Cauchy).
Proof: Here.
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|<= M|Sn - Sn-1|.
So I know you can start with the mean value theorem: f'(x) = (Sn+1 - Sn)/(Sn - Sn-1). Then (Sn - Sn-1)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.