# Composition of Inflationary Mappings is Inflationary

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## 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 $x \in S$.

\(\text {(1)}: \quad\) | \(\displaystyle x\) | \(\preceq\) | \(\displaystyle g \left({x}\right)\) | $g$ is inflationary | |||||||||

\(\text {(2)}: \quad\) | \(\displaystyle g \left({x}\right)\) | \(\preceq\) | \(\displaystyle f \left({g \left({x}\right)}\right)\) | $f$ 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.

$\blacksquare$