Completely Additive Function is Additive

Theorem

Let $f: \N \to \C$ be a completely additive function.

Then $f$ is also additive.

Proof

Let $m, n$ be coprime integers.

Then in particular, $m, n \in \Z$.

Hence, since $f$ is completely additive:

$f \left({m \times n}\right) = f \left({m}\right) f \left({n}\right)$

and $f$ is additive, as desired.

$\blacksquare$