Fixed Point of Mappings is Fixed Point of Composition

Theorem
Let $S$ be a set.

Let $f, g: S \to S$ be mappings.

Let $x \in S$ be a fixed point of both $f$ and $g$.

Then $x$ is a fixed point of $f \circ g$, the composition of $f$ and $g$.

Proof
Since $x$ is a fixed point of $g$:


 * $g(x) = x$

Thus:


 * $(1)\quad f(g(x)) = f(x)$

Since $x$ is a fixed point of $f$:


 * $f(x) = x$

Thus by $(1)$ and the fact that Equality is Transitive:


 * $f(g(x)) = x$

By the definition of composition:


 * $(f \circ g)(x) = x$

Thus $x$ is a fixed point of $f \circ g$.