Let $\displaystyle M$ be a compact metric space and $\displaystyle \Phi:M\longrightarrow M$ be such that $\displaystyle d(\Phi(x),\Phi(y))<d(x,y)$ for all $\displaystyle x,y\in M$, $\displaystyle x\neq y$.

Show that $\displaystyle \Phi$ has a unique fixed point. [Hint: Minimize $\displaystyle d(\Phi(x),x)$.]

I'm not really sure how to use the hint, or even start the problem. This is in the chapter on the contraction mapping principle, so it seems like I have to get it so that I can apply the CMP. Any suggestions would be most welcome. I don't really want the whole problem solved. If you solve the whole problem, at least put some of it in a spoiler, because I'd like to solve as much of this on my own as I can.