Yankees24 said:Great! Could you possibly give me an idea of where to begin with the careful proof? This is usually where I struggle. Thank you!
The supremum (sup) of a set of limit points is the smallest number that is greater than or equal to all of the limit points in the set. The infimum (inf) is the largest number that is less than or equal to all of the limit points in the set.
To find the sup, you need to first identify all of the limit points in the set. Then, you can either use a visual representation, such as a graph, or use algebraic methods to determine the smallest number that is greater than or equal to all of the limit points. Similarly, to find the inf, you would use the same process to determine the largest number that is less than or equal to all of the limit points.
No, a set of limit points can only have one sup and one inf value. This is because the sup and inf are unique and represent the smallest and largest possible values in the set.
The sup and inf of a set of limit points are always related in the following way: sup ≥ inf. In other words, the sup is always greater than or equal to the inf.
The sup and inf of a set of limit points are important concepts in calculus, particularly in the study of limits and continuity. They help to define the behavior of a function at a particular point and can be used to prove the existence or non-existence of a limit. Additionally, the sup and inf are used in the Intermediate Value Theorem, which states that if a function is continuous on a closed interval, then it must take on every value between its sup and inf at least once within that interval.