Composition of Inflationary Mappings is Inflationary

Theorem
Let $\left({S, \preceq}\right)$ be an ordered set.

Let $f, g: S \to S$ be inflationary mappings.

Then $f \circ g$, the composition of $f$ and $g$, is also inflationary.

Proof
Let $x \in S$.

Since this holds for all $x \in S$, $f \circ g$ is inflationary.