Determine Infimum and Supremum of Real Numbers

Determine the infimum and supremum of the real numbers.

Determining the infimum and supremum of a set of real numbers is a fundamental problem in real analysis, reflecting the concepts of bounds and completeness. The infimum, or greatest lower bound, is the largest value that is less than or equal to all the numbers in the set, while the supremum, or least upper bound, is the smallest value greater than or equal to all the numbers in the set. These concepts are central to understanding the completeness property of the real numbers, which states that every non-empty set of real numbers that is bounded above has a supremum and every non-empty set bounded below has an infimum.

When tackling such problems, it's essential to examine the nature of the set of numbers involved. If the set is well-defined and closed, the infimum and supremum could be elements of the set itself. However, with open or infinite sets, these bounds might not be elements of the set but rather limit points. The distinction between maximum, minimum, infimum, and supremum should also be clear: maximum and minimum refer to actual elements within the set, whereas infimum and supremum might not necessarily be within the set.

This exploration not only helps in understanding how bounds work but also sets the stage for further exploration into more advanced topics like sequences, series, and functions where bounds play critical roles in convergence and continuity analyses.