Notice for all and . If there was another number (assume ) such that we would have by Rolle's theorem that there is a point such that which is a contradiction. I don't see where you need to be open though.
A number c is called a fixed point of f if f(x) = c. Prove that if f is differentiable on an interval I and f'(x) < 1 for all x I, then f has at most one fixed point in I. HINT: From g(x) = f(x) - x.
Assum that I is an open interval (why must we assume this?)