I'm having trouble with this problem. Could someone give me a hand?
Let S be a nonempty subset of R that is bounded above. Show that if sup S is not in S, then for every , the interval contains infinitely many elements of S.
I'm having trouble with this problem. Could someone give me a hand?
Let S be a nonempty subset of R that is bounded above. Show that if sup S is not in S, then for every , the interval contains infinitely many elements of S.
Suppose for some that contains only finitely many points of (that it is non-empty should be obvious).
Thanks a lot for your help, CB. I actually tried to assume the contrary as you suggested, but I got stuck. I have is in . So, I think I should be able to show that this lead to sup S is in S, but I don't know how.
Thanks a lot for your help, CB. I actually tried to assume the contrary as you suggested, but I got stuck. I have is in . So, I think I should be able to show that this lead to sup S is in S, but I don't know how.
A finite subset of the reals has a largest element, which will be an upper bound for and less than which is a contradiction etc...