Let $F,G$ to be distribution. Is the product $FG$ also a distribution? 
Let $F,G$ to be distribution functions of a random variable, say $X$. Is the product $FG$ also a distribution?

We define $\psi(x) = F(x)G(x)$. We first have:
$$\lim_{x \to \infty } \psi(x) = \lim_{x \to \infty} F(x)G(x) = 1,$$
$$\lim_{x \to -\infty } \psi(x) = \lim_{x \to -\infty} F(x)G(x) = 0.$$
because both $F,G$ are distribution. Also, if $x < y$, then 
$$\psi(x) \leq \psi(y) \Leftrightarrow F(x)G(x) \leq F(y)G(y),$$
but $F(x) \leq F(y)$ and $G(x) \leq G(y)$, so the previous inequality holds. Finally, we have
$$\lim_{h \to 0+ } \psi(x + h) = \lim_{h \to 0+ }F(x+h)G(x+h) = F(x)G(x) = \psi(x).$$
Is this reasoning correct? I am not sure, if the products of two functions is defined I used them. I am also not sure if the operations make sense. 
Thanks in advance

Your reasonment is correct, your function $FG$ matches all criteria of a CDF:


*

*increasing

*is $0$ in $-\infty$ and $1$ in $\infty$

*defined on $\mathbb{R}$
Interestingly, $FG$ is the cumulative distribution function of $max(X_F,X_G)$ if $(X_F,X_G)$ are independent random variables with F and G as CDF :
$\forall x\in\mathbb{R} \quad \mathbb{P}(max(X_F,X_G)\leq x)=\mathbb{P}(X_F\leq x ,X_G \leq x )=\mathbb{P}(X_F\leq x )\mathbb{P}(X_G \leq x )=F(x)G(x)$
