Setup:
A set of random variables $X_1, \ldots X_n$ is independent if for all Borel-sets $B_1,\ldots,B_n$ it is the case that $$P(X_1 \in B_1,\ldots,X_n \in B_n) = \prod_{i=1}^n P(X_i \in B_i)$$
An alternative definition is that the generated sigma-algebras $\sigma(X_1),\ldots,\sigma(X_n)$ should be independent.
Question:
Let $X, Y, Z$ be independent random variables. I am pretty sure that this implies that $XY$ and $Z$ are also independent, but I have not been able to prove this.
Is there an elementary way to prove this or is there a trick or a general theorem needed?
Thank you.