I need help to know (in detail) how to prove that the product of two multiplicative arithmetic functions is a multiplicative arithmetic function.
$$$$$f(n)$ and $g(n)$ are functions multiplicative, show that $$h(n)=f(n)\cdot g(n)$$is multiplicative function.
If it is the simple product it's a one liner as Andre says:
$$fg(mn):=f(mn)g(mn)=f(m)f(n)g(m)g(n)=f(m)g(m)f(n)g(n)=fg(m)\cdot fg(n)$$