Definition 3

Let f be a real-valued function defined on a set X, and let A be a non-empty subset of X. Then the least upper bound (resp. greatest lower bound) of the set $f(A)$ in $\overline{\mathbf{R}}$ is called the least upper bound (resp. greatest lower bound) of f in A, and is denoted by

In particular, if A is a non-empty subset of $\overline{\mathbf{R}}$, then

It is often more convenient to use the notation on the right-hand side to denote the least upper bound of A.

The number $a = \sup_{x \in A} f(x)$ is characterized by the following two properties: (i) For all $x \in A, f(x) \leq a$. (ii) For every $b < a$, there exists $x \in A$ such that $b < f(x) \leq a$.

The numbers $\sup_{x \in A} f(x)$ and $\inf_{x \in A} f(x)$ belong to the closure of $f(A)$ in $\overline{\mathbf{R}}$. We have $\inf_{x \in A} f(x) \leq \sup_{x \in A} f(x)$; and these two numbers are equal if and only if $f$ is constant on A.

A real-valued function, defined on a set X, is bounded above (resp. bounded below) in a non-empty subset A of X if and only if $\sup_{x \in A} f(x) < + \infty$ [resp. $\inf_{x \in A} f(x) > -\infty$]. $f$ is bounded in A if and only if $|f|$ is bounded above in A, hence if and only if $\sup_{x \in A} |f(x)| < + \infty$.

We have

This relation reduces all properties of the greatest lower bound to those of the least upper bound; hence in general we shall speak only of the latter.