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The infimum of a set is the greatest lower bound, meaning that it is the largest number that is still less than or equal to all the numbers in the set. The supremum of a set is the least upper bound, meaning that it is the smallest number that is still greater than or equal to all the numbers in the set.
Infimum and supremum are important concepts in mathematics because they help us define and understand the boundaries of a set. They also allow us to prove the existence of certain elements in a set and make precise statements about the behavior of functions and sequences.
To prove a problem involving infimum and supremum, you need to use the definitions of these concepts and the properties of real numbers. You may also need to use mathematical induction or other proof techniques to establish the existence of certain elements in the set and the validity of the given statement.
One example of a problem on infimum and supremum is proving that the infimum of the set {(−1)n + 1/n | n ∈ N} is -2 and the supremum is 2. This can be proven by showing that -2 is the greatest lower bound and 2 is the least upper bound, and then using mathematical induction to prove that these bounds are indeed the infimum and supremum of the set.
Infimum and supremum have many real-world applications, especially in economics, physics, and engineering. They can help us determine the minimum and maximum values of a function, optimize resources, and predict the behavior of systems. For example, in economics, infimum and supremum can be used to find the most profitable price for a product, while in physics, they can help determine the maximum and minimum possible values of physical quantities.