The set $\mathbb R\cup \{-\infty,+\infty\},$ i.e the set of real numbers extended by the improper real numbers $-\infty$ minus infinity) and $+\infty$ plus infinity), is called the extended real numbers and denoted by $\overline{\mathbb R}.$
Extending the order relation for real numbers, we define $-\infty < x < +\infty$ for all $x\in\mathbb R.$
Definitions: 1 2 3 4 5
Lemmas: 6 7
