Composition of Inflationary Mappings is Inflationary

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Theorem

Let $\struct {S, \preceq}$ 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 $x \in S$.

\(\text {(1)}: \quad\) \(\ds x\) \(\preceq\) \(\ds \map g x\) $g$ is inflationary
\(\text {(2)}: \quad\) \(\ds \map g x\) \(\preceq\) \(\ds \map f {\map g x}\) $f$ is inflationary
\(\ds x\) \(\preceq\) \(\ds \map f {\map g x}\) $(1)$ and $(2)$ and $\preceq$ is an ordering and hence transitive
\(\ds \leadsto \ \ \) \(\ds x\) \(\preceq\) \(\ds \map {\paren {f \circ g} } x\) Definition of Composition of Mappings

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

$\blacksquare$