[Contraction mapping theorem; cf. the last problem of the Topology IV set.] A function f from a metric space X to itself is said to be a contraction if there exists a constant c<1 such that d(f(x),f(y))≤cd(x,y) for all x,y∈X [i.e., f shrinks all distances by a factor of at least 1:c]. Prove that f is continuous, and that it has at most one fixed point, i.e., there is at most one z∈X such that f(z)=z. Give an example of a nonempty X and a contraction map on X without a fixed point. Show that if X is nonempty and complete then every contraction map has a (necessarily unique) fixed point. Get tutor help
