Let f : [0,) R be a continuously differentiable function such that f(0) > 0 and there is a constant C with |f 0 (x)| C < 1 for all x.

(a) Show that f(x) f(0) + Cx for all x > 0.

(b) Prove that limx[f(x) x] = .

(c) Show that there is a point x0 > 0 such that f(x0) = x0.

(d) Prove that there is only one point x0 > 0 satisfying f(x0) = x0.

I am lost on where to begin for all of these. If someone can start each proof for me, that would be greatly appreciated.