I just learned about these yesterday, so I'm having a little trouble with this problem.

If T: [0,1] to [0,1] and if there is a real number alpha with 0<= alpha < 1 such that |T'(x)| <= alpha (0 <= x <= 1) where T' is the derivative of T, prove that T is a contraction on [0,1].

Any help would be appreciated!