2. Infimum and supremum

A number x\in\mathbb{R} is an upper bound of a set A\subseteq\mathbb{R} if for every a\in A we get a\leq x. Similarly x\in\mathbb{R} is a lower bound of A\subseteq\mathbb{R} if for every a\in A we get a\geq x.

If x is an upper bound of A and additionally x\in A it will be called the greatest element of A, and such a lower bound will be called the least element of a set. Obviously a set may not have greatest or least element, eg. in (0,1) there is no least element and neither there is the greatest one.

The least upper bound of a set A (if it exists) will be called the supremum of A and denoted as \sup A. The greatest lower bound of A (if it exists) will be called infimum of A and denoted \inf A. Notice that sometimes supremum or infimum may not be an element of the given set, eg. \sup(0,1)=1\notin (0,1).

Other example. Let B=\left\{\frac{1}{n+1}\colon n\in\mathbb{N}\right\}. Then \sup B=1, because it is the greatest element in this set — for every n\in\mathbb{N}, \frac{1}{n+1}\leq 1 and \frac{1}{0+1}=1. Now, \inf B=0. Indeed, for all n\in\mathbb{N} we get \frac{1}{n}\geq 0, because 1>0 (so 0 is a lower bound of B). Moreover, it is the greatest lower bound, because if x>0, let n be a natural number greater than \frac{1}{x}. Then \frac{1}{n+1}<x, so x is not a lower bound of B. We have proved that 0 is its infimum.

If a set A has no upper bound we will write \sup A=\infty and if it has no lower bound, we will write \inf A=-\infty.