I need help this.
Let (X,d) be a compact metric space and f:X goes to X be a function satisfying for every x,y element X x does not equal y implies d(f(x), f(y)) < d(x,y)
also phi(x) = d(x,f(x)) is uniformly continuous for x element X
Show that f has a fixed point i.e. there is an x0 element of X f(x0)=x0
do I need the Banach Contraction Principle, fixed point theorem, or intermediate value theorem?