# Differentiation Analysis

• November 26th 2007, 03:33 PM
tbyou87
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|<= 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.
• November 26th 2007, 06:01 PM
ThePerfectHacker
Theorem: If $0 and $|s_{n+1}-s_n| \leq a|s_n - s_{n-1}|$. Then the sequence is convergent (or Cauchy).

Proof: Here.