I have based the following on a proof in Xavier Vives' book 'Oligopoly Pricing'. The proof is simple and elegant.
Theorem. Let f:[0,1]->[0,1] where f is increasing. Then f has a fixed point. (Note that we are not assuming that f is continuous.)
Proof. Let X={x in [0,1]:f(x)>=x}. Assume 0 is not in X. Then f(0)<0 contradicting the definition of X. So X is non-empty and its least upper bound, u, exists. Also u is in [0,1]. Let x be in X. Then u>=x and since f is increasing f(u)>=f(x)>=x. So f(u) is an upper bound of X and f(u)>=u by the definition of u. Now f(u) is in [0,1] and since f is increasing f(f(u))>=f(u) so f(u) is in X and hence u>=f(u). Thus f(u)=u and we are done.
Try the same thing with the set Y={x in [0,1]:x>=f(x)} and the greatest lower bound.