Define the supremum or least upper bound of a non-empty set of real numbers.

Let $A$ denote a non-empty set of real numbers which has a supremum but no maximum. Show that for every $\epsilon>0$ there are infinitely many elements of $A$ contained in the open interval

$$(\sup A-\epsilon, \sup A) .$$

Give an example of a non-empty set of real numbers which has a supremum and maximum and for which the above conclusion does not hold.