infimum and supremum
Infimum and supremum
First we'll go through some rudimentary definitions.
A set \[A\subset\mathbb{R}\] (\[A\] is a subset of real numbers) is bounded from above if there exists a real number \[M\] (or \[M\in\mathbb{R}\]), known as the upper bound of \[A\], such that \[x\le M\] for every \[x\in A\]. Simply speaking, a set (\[A\]) is bounded above (\[M\]) if there is a real number that serves as a ceiling, beyond which no elements of the set can go. Similarly, \[A\] is bounded from below if there exists \[m\in\mathbb{R}\], known as the lower bound of \[A\], such that \[x\ge m\] for every \[x\in A\]. We can then say that a set is bounded if it is bounded both from above and below. Keep in mind that infinity is not a real number.
Thus, the infimum of a set is its greatest lower bound, sometimes seen as \[\text{glb}\] or \[\inf\], and it would be the largest real number that is smaller or equal to every element in set \[A\]. On the other hand, the supremum of a set is its least upper bound (\[\text{lub}\] or \[\sup\]), or the smallest real number that is larger or equal to every element in set \[A\].
Consider the set \[A=\left\{ x\in\mathbb{R} \mid x>0\right\}\]. The infimum of this set, according to the definition would be 0, as 0 is the largest real number that is less than or equal to every element in \[A\]. The supremum of this set does not exist, as the set is not bounded above (the set extends infinitely, infinity is not a real number, thus we cannot find a real number that bounds this set).
Some people may be inclined to think that the supremum and infimum is equivalent to the min/max that is usually used, it is similar, but not exactly the same. These concept of infimum and supremum is more useful in analysis because they better characterize special sets which may have no minimum or maximum such as the set \[A\] that we just defined above.
Consider the set \[A=\left\{ x\in\mathbb{R}\mid-2<x<5 \right\}\]. We can't really define a min/max of this set, e.g. setting the max to 5 would be incorrect as 5 does not exist in the set, while setting the max to 4.99 would be also incorrect as the number 4.9999999 exists in the set and so on. However, we would be able to define the infimum to be -2 and the supremum to 5.
Definition
Suppose that \[A\subset\mathbb{R}\] is a set of real numbers.
If \[M\in\mathbb{R}\] is an upper bound of \[A\] such that \[M\le M^{\prime}\] for every upper bound \[M^{\prime}\] of \[A\], then \[M\] is called the supremum of set \[A\], denoted \[M=\sup{A}\]. In simple words, \[M\] is the supremum if and only if it is smaller or equal to every single possible upper bound of \[A\], thus the name lowest upper bound (referencing the earlier example, valid upper bounds of set \[A\] can be 5, 6 or even 10000). If \[m\in\mathbb{R}\] is a lower bound of \[A\] such that \[m\ge m^{\prime}\] for every lower bound \[m^{\prime}\] of \[A\], then \[m\] is called the infimum of of \[A\], denoted as \[m=\inf A\].
- If \[A\] is not bounded from above, then we write \[\sup A=\infty\], and if it isn't bounded from below, then \[\inf A=-\infty\].
- If \[A\] is an empty set, \[A=\emptyset\], then every real number is both an upper and a lower bound of \[A\], and we write \[\sup \emptyset=-\infty\] and \[\inf\emptyset=\infty\].