Let S be a nonempty set of real numbers that is bounded above, and let Beta be the least upper bound of S. Prove that for every Epsilon greater than 0, there exists an element x such that x is greater than Beta minus Epsilon.
Any ideas or help would be appreciated. Thanks.
The statement that is not an upper bound of S means that . That is some number in S is strictly greater that .
The statement that is the least upper bound of S means that is an upper bound of S no number less than is an upper bound.
If then .
Now you put those three facts together to form a proof of the proposition.