Thanks a million. Still now too sure on your answer after the bounded part.
This is my full question that I have to answer -
Let X=[1,infinity) with the metric d(x,y) = |x-y|. Define T:X->X by T(x) = x+1/x. Prove T satisfies
d(T(x),T(y)) < d(x,y) for all x,y E X
but T has no fixed points in X. Does this contradict Banach's fixed point theorem.