Composition of Inflationary Mappings is Inflationary

Theorem
Let $(S, \preceq)$ 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 $x \in S$.

$x \preceq g(x)$ because $g$ is inflationary.

$g(x) \preceq f(g(x))$ because $f$ is inflationary.

Thus $x \preceq f(g(x))$ by transitivity.

Therefore $x \preceq (f \circ g)(x)$ by the definition of composition.

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