Positive Infinity is Greatest Element

Theorem

Let $\left({\overline \R, \le}\right)$ be the extended real numbers with their usual ordering.

Then $+\infty$ is the greatest element of $\overline \R$.

Proof

We have, by definition of the usual ordering on $\overline \R$:

$\forall x \in \overline \R: x \le +\infty$

That is, $+\infty$ is the greatest element of $\overline \R$.

$\blacksquare$

Also see

• Negative Infinity is Smallest Element