1. The problem statement, all variables and given/known data Let X be a compact metric space. if f:X-->X is continous and d(f(x),f(y))<d(x,y) for all x,y in X, prove f has a fixed point. 2. Relevant equations 3. The attempt at a solution Assume f does not have a fixed point. By I problem I proved before if f is continous with no fixed point then there exists an epsilon>0 st d(f(x),x)>=epsilon for all x in X. Using this I wanted to get a contradiction. i wanted to prove d(f(x),f(y))>=d(x,y) which leads to a contradiction of d(f(x),f(y))<d(x,y) but I got stuck.