Suppose f : R R is differentiable and there exists a constant C 1 so that |f |f ' (x)| C for all x R. Prove that f has a fixed point, i. there...

Suppose f : R → R is differentiable and there exists a constant C < 1 so that |f |f ' (x)| ≤ C for all x ∈ R. Prove that f has a fixed point, i.e. there exists at least one x ∈ R so that f(x) = x. (Hint: pick an arbitrary x1 and define a sequence xn recursively by xn+1 = f(xn).)