Proving Contraction Constants: Mean Value Theorem Help | Homework Example

[Easty]

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?

 

[HallsofIvy]

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.

 

[Easty]

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?