Strictly Monotone Mapping is Monotone

Theorem

A mapping that is strictly monotone is a monotone mapping.

Proof

If $\phi$ is strictly monotone, then it is either strictly increasing or strictly decreasing.

If $\phi$ is strictly increasing, then by Strictly Increasing Mapping is Increasing, $\phi$ is increasing.

If $\phi$ is strictly decreasing, then by Strictly Decreasing Mapping is Decreasing, $\phi$ is decreasing.

Thus $\phi$ is monotone by definition.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings