Infimum

The concepts of infimum and supremum are close to minimum and maximum, but are more useful in analysis because they better characterize special sets which may have no minimum or maximum. For instance, the set of positive real numbers${\displaystyle \mathbb {R} ^{+}}$ (not including ${\displaystyle 0}$) does not have a minimum, because any given element of ${\displaystyle \mathbb {R} ^{+}}$ could simply be divided in half resulting in a smaller number that is still in ${\displaystyle \mathbb {R} ^{+}.}$ There is, however, exactly one infimum of the positive real numbers relative to the real numbers: ${\displaystyle 0,}$ which is smaller than all the positive real numbers and greater than any other real number which could be used as a lower bound. An infimum of a set is always and only defined relative to a superset of the set in question. For example, there is no infimum of the positive real numbers inside the positive real numbers (as their own superset), nor any infimum of the positive real numbers inside the complex numbers with positive real part.