Positive Infinity is Greatest Element
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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 $\overline \R$.
$\blacksquare$