Mean value theorem help

Homework Statement

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.

Homework Equations

|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?

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HallsofIvy
Homework Helper

I don't see why you should use derivatives at all. If $h(x)= h_2(h_1(x))$ then $|h(a)- h(b)|= |h_2(h_1(a))- h_2(h_1(b))|\le \delta_2|h_1(a)- h_1(b)|$ and repeat.

I don't see why you should use derivatives at all. If $h(x)= h_2(h_1(x))$ then $|h(a)- h(b)|= |h_2(h_1(a))- h_2(h_1(b))|\le \delta_2|h_1(a)- h_1(b)|$ and repeat.

but then arent you assumeing that h(x)= h_2(h_1(x)) is a contration?