If h1 and h2 are contractions on a set B with contraction constants δ1 and δ2 prove that the composite function h2 ° h 1 is also a contraction on B and find a contraction constant.
|f(a) - f(b)| ≤ δ |a-b|
f '(c) = (f(a)-f(b))/(a-b)
(g°f) '(c) = g '(f(c))x f '(c)
The Attempt at a Solution
So far i'm pretty sure i have to use the mean value theorem and the chain rule. using the mean value on the composite function i get :
|h2(h1(a)) - h2(h1(b))| = |h2 °h1 )' (c)| |a-b|
i get stuck here, i think i should now use the chain rule for the derivaitve term out the front of the equality to somehow make an inequality. am i on the right track?