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Infimum is a mathematical concept that refers to the greatest lower bound of a set of numbers. It is a fundamental idea in the study of real numbers and is used extensively in analysis and calculus. The infimum of a set S, denoted as inf(S), is the largest number that is less than or equal to every element in S. If no such number exists, the infimum is said to be negative infinity.

The concept of infimum is closely related to that of supremum, which is the least upper bound of a set. Together, infimum and supremum provide a way to describe the bounds of a set of numbers. For example, if S is the set of all rational numbers between 0 and 1, then inf(S) = 0 and sup(S) = 1.

Infimum is particularly useful in the study of sequences and series. For instance, if {a_n} is a sequence of real numbers, the infimum of the sequence is the greatest lower bound of the set of all terms in the sequence. This concept is used to define the limit inferior of a sequence, which is the limit of the infimum of the sequence as the index approaches infinity.

In summary, infimum is a crucial concept in mathematics that helps to describe the lower bounds of a set of numbers. It is used extensively in analysis and calculus, and is closely related to the concept of supremum.