Product of Multiplicative Functions is Multiplicative

Theorem
Let $f: \N \to \C$ and $g: \N \to \C$ be multiplicative functions.

Then their pointwise product:
 * $f \times g: \Z \to \Z: \forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$

is also multiplicative.

Proof
Let $f$ and $g$ be multiplicative.

Let $m \perp n$.

Then:

Hence the result by definition of multiplicative function.