Definition 1.2.1. \(\overline{\mathbb R}\) is the set \(\mathbb R\) with two distinct sets \(-\infty\) and \(+\infty\) which are not belong to \(\mathbb R.\) Equivalently,
\[\overline{\mathbb R} = \mathbb R \cup \{-\infty, +\infty\}.\]Then we extend the usual order on \(\mathbb R\) to observe the following rule:
\[-\infty < x < +\infty\]for all \(x\in\mathbb R.\)
Definition 1.2.2. Let \(A \subseteq \mathbb R\). Its extended supremum and extended infimum are defined by
\[\sup_{\overline{\mathbb R}}A = \begin{cases} \sup A, & \text{if } A \text{ is nonempty and bounded above}, \\ +\infty, & \text{if } A \text{ is unbounded above}, \\ -\infty, & \text{if } A = \varnothing, \end{cases}\]and
\[\inf_{\overline{\mathbb R}}A = \begin{cases} \inf A, & \text{if } A \text{ is nonempty and bounded below}, \\ -\infty, & \text{if } A \text{ is unbounded below}, \\ +\infty, & \text{if } A = \varnothing. \end{cases}\]Definition 1.2.3. For all \(x\in \mathbb R,\)
\[x + (+\infty) = +\infty, \qquad x + (-\infty) = -\infty, \qquad \frac{x}{+\infty} = \frac{x}{-\infty}=0.\]If \(0<x\) then
\[x\cdot (+\infty) = +\infty, \qquad x\cdot (-\infty) = -\infty,\]and, if \(x<0\) then
\[x\cdot (+\infty) = -\infty, \qquad x\cdot (-\infty) = +\infty.\]Exercises
References
- Rudin, W. (1976). Principles of mathematical analysis (3rd ed.). McGraw-Hill Education.