I'm having a lot of trouble with contractions and open coverings for my real analysis class, and I was hoping someone would be able to help me with a few of the problems:
1) Give an example of a continuous bounded function on (-inf, inf) that attains neither a maximum nor minimum value
2a) For all x in (0,1), let I_x denote the open interval (x/2, (x+1)/2). Show that the family G of all such I_x is an open covering of (0,1) which admits no finite subcovering of (0,1)
2b) Add two appropriate sets to the family G (from above) to form an open covering H of [0,1]. Show that H does admit a finite subcovering of [0,1].