Suppose is such that then is is continous. is compact. If in attains a minimum, what can you say about ?
Let be a compact metric space and be such that for all , .
Show that has a unique fixed point. [Hint: Minimize .]
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.