1. The problem is to show that the set $ (1,2) $ has no maximum element. 2. The set $ (1,2) $ is an open interval, meaning it includes all real numbers between 1 and 2 but does not include the endpoints 1 and 2 themselves. 3. To have a maximum, the set must contain a largest element $ m $ such that for all $ x $ in the set, $ x \leq m $ and $ m $ is in the set. 4. Let's suppose that the set $ (1,2) $ does have a maximum $ m $. 5. Since $ m $ must be in the set, $ 1 < m < 2 $. 6. But because $ (1,2) $ is dense in the real numbers, we can always find a number $ m' $ such that $ m < m' < 2 $. 7. This $ m' $ would also be in $ (1,2) $ and clearly greater than $ m $, contradicting the assumption that $ m $ is the maximum. 8. Therefore, the set $ (1,2) $ has no maximum element.