0.7 The Extended Real Number System

We briefly talk about materials in elementary real analysis. We will not prove those properties (see elementary real analysis textbook such as Principles of Mathematical Analysis for reference).

We form the extended real number system by adding two elements $-\infty$ and $\infty$ on $\mathbb{R}$; $$ \bar{\mathbb{R}} = \mathbb{R} \cup \left\{ -\infty, \infty \right\} $$ Also we extend the usual ordering $\leq$ on $\mathbb{R}$ by defining $$ -\infty < x < \infty, (\forall x \in \mathbb{R}) $$