Proof for interval within contraction mapping theorem

Question

So, when proving the contraction mapping theorem this book that I am using seemed to imply that if g is a contraction on [a,b], then if we take any two points inside this interval say [c,d], then g is also a contraction on c and d. Now if g is monotonically increasing, I can see why this is true, but otherwise, I just don't understand how this can be true ALL THE TIME.

For instance, say we have some function such that it is indeed a contraction on [a,b] but it may not be a contraction on s a very small interval say [c,d] which is within [a,b].

Can someone please help?

Answer 1

Suppose $|f(x)-f(y)| \leq c |x-y|$ for all $x, y \in [a,b]$ where $c <1$. Then the same inequality holds for $x,y \in [c,d]$ if $[c,d] \subseteq [a,b]$. Why is monotonicity required here?