lowerboundedness

Lowerboundedness is a concept used in mathematics, particularly in set theory and analysis, to describe a property of a set or a function. A set of numbers is considered lower bounded if there exists a number that is less than or equal to every element in the set. This number is called a lower bound. For example, the set {3, 5, 7, 9} is lower bounded because 3 is less than or equal to all its elements. Even 0 or -10 would also be valid lower bounds for this set.

A function is lower bounded if its range is a lower bounded set. This means there is a real number M such that for all values in the domain of the function, the function's output is greater than or equal to M. For instance, the function f(x) = x^2 is lower bounded because for any real number x, x^2 is always greater than or equal to 0. Therefore, 0 is a lower bound for this function.

The greatest of all lower bounds for a set is called the infimum of the set. If a set has a lower bound, it is said to be bounded below. Conversely, a set that is not lower bounded is said to be unbounded below. For example, the set of all integers is unbounded below as there is no smallest integer. The concept of lowerboundedness is crucial in optimization problems, analysis of sequences and series, and the study of mathematical spaces.