Extended Real Numbers

Definition

We define the extended real number line to be

$$\mathbb{R}\cup \{-\infty ,\infty \}$$

1. If $A\subset \mathbb{R}$ and bounded, then $\sup (A),\inf (A)$ is defined as before.
2. If $A\subset \mathbb{R}$ and not bounded from above, then $\sup A=\infty$
3. If $A\subset \mathbb{R}$ and not bounded below, then $\inf A=-\infty$
4. If $A$ includes $\infty$, then $\sup A=\infty$ (and likewise with $-\infty$)
5. If $A=\{\infty \}$, then $\inf A=\sup A=\infty$ (and likewise with $-\infty$)