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 also a fixed point of $f \circ g$, the composition of $f$ and $g$.

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


 * $g \left({x}\right) = x$

Thus:


 * $f \left({g \left({x}\right)}\right) = f \left({x}\right)$

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


 * $f \left({x}\right) = x$

It follows that:


 * $\left({f \circ g}\right) \left({x}\right) = f \left({g \left({x}\right)}\right) = x$

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