Composition of Increasing Mappings is Increasing

Theorem
Let $\struct {S, \preceq_S}$, $\struct {T, \preceq_T}$ and $\struct {U, \preceq_U}$ be ordered sets.

Let $g: S \to T$ and $f: T \to U$ be increasing mappings.

Then their composition $f \circ g: S \to U$ is also increasing.

Proof
Let $x, y \in S$ with $x \preceq_S y$.

Since $g$ is increasing:


 * $\map g x \preceq_T \map g y$

Since $f$ is increasing:


 * $\map f {\map g x} \preceq_U \map f {\map g y}$

By the definition of composition:


 * $\map {\paren {f \circ g} } x \preceq_U \map {\paren {f \circ g} } y$

Since this holds for all such $x$ and $y$, $f \circ g$ is increasing.