Assume $f$ has a finite derivative and $|f'(x)| \leq y < 1$ for all $x \in (a,b)$
$f$ is continuous and $a \leq f(x) \leq b$ for all $x \in [a,b]$. Prove $f$ has a unique fixed point in $[a,b]$.
So far I have for every c in (a,b) |f'(c)| ≤ y
=> lim x->c |f(x) - f(c)|/|x-c| ≤ y
=> lim x->c |f(x) - f(c)| ≤ y lim x->c |x-c|
Would that be the definition of a contractive map in R?
Therefore by Banach Fixed Point Theorem, f has a unique fixed point.
Can I prove Banach's theorem using the mean value theorem?