Okay guys, I have this question that troubles me a lot. Is there an example of a function that is continuous on a closed and bounded set but achieves no maxima?
My take is, that apparently cannot be in Euclidean space. I think of bounded sequences, L-inf'ty, where seqs are bounded and closed (all limits contained) but my puzzle is then why not to have a maximum (i.e. subseqs will not also converge to the limit contained)? Plus that I cannot come up with an example of such a function.
Thanks a lot.
P.S. THis is my first post and I am a rookie in math, so I apologise if I don't express something very clearly.
Edit: Thanks a lot for your super quick responses! I forgot to mention that I consider a sup metric in L-inf'ty space I mention. Of course this is just an example.