If $(X,d)$ is a complete metric space and $\xi:\;X\to X$ satisfies:
$$d(x,y)<n+1\Rightarrow d(\xi(x),\xi(y))<n$$$$d(x,y)<1/n\Rightarrow d(\xi(x),\xi(y))<1/(n+1)$$for all $n= 1,2,\dots$, does $\xi$ necessarily have a fixed point?
I am not sure what to do with this question; I would guess the answer is no, but I am having trouble constructing a counter-example. If someone could offer a hint I would be thankful.