If it does, is it open? What say you?
There must be a defintion of "Open Interval" sitting about somewhere. Why not have a good, close look at it?
Indirect proof: Suppose the open interval, (a, b), does have a maximum, M. Since M is in (a, b), a< M< b. Let .
1) N< b.
2) a< M< N
so N is in (a, b)
3) M< N, contradicting the hypothesis.
Do you understand the difference between a "maximum" and a "supremum" (least upper bound)? That is crucial here.