  Let $f\colon\mathbb{N}\rightarrow\mathbb{N}_0$ be a multiplicative arithmetic
function such that for all primes $p$ and positive integers $\alpha$,
$f(p^{\alpha})<p^{\alpha}$ and $f(p)\vert f(p^{\alpha})$. Suppose also that any
prime that divides $f(p^{\alpha})$ also divides $pf(p)$. Define $f(0)=0$, and
let $H(n)=\displaystyle{\lim_{m\rightarrow\infty}f^m(n)}$, where $f^m$ denotes
the $m^{th}$ iterate of $f$. We prove that the function $H$ is completely
multiplicative.
