Proving inf(A)=0 means showing that the infimum (or greatest lower bound) of a set of numbers A is equal to 0. In other words, there is no number smaller than 0 in the set A.
Proving inf(A)=0 is important because it helps us understand the behavior and properties of a set of numbers. It also allows us to make certain conclusions and predictions about the set and its elements.
To prove inf(A)=0, you need to show that 0 is the greatest lower bound of the set A. This can be done by showing that 0 is less than or equal to all elements in the set, and that no number smaller than 0 can be a lower bound of the set.
One example could be proving the infimum of a set of temperatures is equal to 0, meaning there is no temperature lower than 0 degrees. Another example could be proving the infimum of a set of stock prices is equal to 0, indicating there is no stock price lower than 0 dollars.
Yes, inf(A) can be equal to a number other than 0. It is possible for a set of numbers to have a different greatest lower bound, or infimum, than 0. However, proving inf(A)=0 is a specific task and may require different methods than proving inf(A) = a different number.