The extended real number system

Consists of The real field and two symbols, $+\infty$ and $-\infty$. We preserve the original order in $\mathbb{R}$ and define

$$-\infty <x<+\infty$$

for every $x\in \mathbb{R}$.

$+\infty$ is an upper bound for every subset of the extended real number system, so every non empty subset has an upper bound. Ditto for $-\infty$.

The extended real number system does not form a field, since $\pm \infty$ do not have additive or multiplicative inverses.