(Disclaimer: This is a homework question.)
Prove that if a subset S of $\mathbb{R}$ contains an upper bound, then this upper bound coincides with the supremum of S.
I may be terribly misunderstanding this question. But it seems to me that the upper bound of a subset of $\mathbb{R}$ is somewhat arbitrary. For example, say we look at S = $\left [0,1 \right ]$ $\subset$ $\mathbb{R}$. S is bounded above by any r $\in$ $\mathbb{R}$ with r$\geq$1. But the supremum of S is obviously 1. There doesn't seem to be any reason these two should be equal, unless I'm misunderstanding the problem.
I've attempted to solve problem by using proof by contradiction:
S has a upperbound, that is $\exists$ U $\in$ $\mathbb{R}$ : U $\geq$ s ,$\forall$ s $\in$S.
let M = Sup(S).
The supremum is the least upper bound, that is M $\leq$ U.
Claim M = U.
Otherwise M < U $\implies$ $\exists$ x $\in$ $\mathbb{R}$ such that M $\lt$ x < U.
From here we can try to show that x $\in$ S, which contradicts the supremum property of M, which would imply M = U.
But I can't seem to accomplish that.