Let S be a nonempty set of real numbers bounded from above and let x=supS. Prove x either belongs to the set or is an accumulation point of S.
x is an accumulation point of S iff each neighborhood of x contains a member of S different from x. That is, every neighborhood of x contains infinitely many points of S.
The Attempt at a Solution
So, do I need to prove that if x is not a member of S, then it is an accumulation point? I am a little confused about how to go about this.
So, there would obviously be two possibilities. Either x is a member of S, or it is not. If not, we need to prove x is an accumulation point. Wouldn't we need to know that S is infinite though? Is so, wouldn't it be much like the proof of the Bolzano-Weierstrass theorem?
Any help would be greatly appreciated! Thanks!