The supremum of a set is the smallest number that is greater than or equal to all of the numbers in that set. It is also known as the least upper bound.
The supremum is similar to the maximum, but it may not exist for all sets. If a set has a maximum, then that number is also the supremum. However, if a set does not have a maximum, the supremum can still exist.
Yes, the set of all real numbers less than 1 has a supremum of 1, but it does not have a maximum. This is because there is no single number in the set that is greater than or equal to all of the numbers in the set.
Supremum is a fundamental concept in real analysis and is used to define important properties of sets, such as boundedness and completeness. It is also used in theorems and proofs to establish the existence of certain numbers or limits.
The supremum is the smallest number that is greater than or equal to all of the numbers in a set, while the infimum is the largest number that is less than or equal to all of the numbers in a set. They are essentially the same concept, but with different directions.