# 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 $x \in S$.

 $(1):\quad$ $\displaystyle x$ $\preceq$ $\displaystyle g \left({x}\right)$ $g$ is inflationary $(2):\quad$ $\displaystyle g \left({x}\right)$ $\preceq$ $\displaystyle f \left({g \left({x}\right)}\right)$ $f$ is inflationary $\displaystyle x$ $\preceq$ $\displaystyle f \left({g \left({x}\right)}\right)$ $(1)$ and $(2)$ and $\preceq$ is an ordering and hence transitive $\displaystyle \implies \ \$ $\displaystyle x$ $\preceq$ $\displaystyle \left({f \circ g}\right) \left({x}\right)$ Definition of composition

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

$\blacksquare$