Give an example of a bounded non-empty set of real numbers which hasa supremum, but does not have a maximum. Justify your answercarefully.

An example of a bounded non-empty set of real numbers which has a supremum but does not have a maximum is the open interval (0,1) on the real number line.

Step 1: Define the set Let's define the set S = {x ∈ R : 0 < x < 1}. This is the set of all real numbers between 0 and 1, not including 0 and 1 themselves.

Step 2: Show the set is bounded The set S is clearly bounded below by 0 and above by 1.

Step 3: Show the set has a supremum The supremum of a set is the least upper bound of the set. In this case, 1 is an upper bound of the set S, and there is no number less than 1 that is also an upper bound of S. Therefore, 1 is the supremum of S.

Step 4: Show the set does not have a maximum The maximum of a set is the greatest element in the set. However, for any element x in S, we can always find another element that is greater than x but still less than 1 (for example, x + (1 - x) / 2). Therefore, S does not have a maximum.

This example shows that it is possible for a set to have a supremum but not a maximum. The key is that the supremum is an upper bound of the set, but it does not have to be an element of the set itself.