I'm stuck with this exercise...
Let $\{X_n\}$, $\{Y_n\}$ r.v. bounded and independent such that $$\frac{1}{N} \sum_{n=1}^{N}X_n \to p \ \ \ \text{a.s.},$$ prove that $$\frac{1}{N}\sum_{n=1}^{N}X_n Y_n \to pZ \ \ \ \text{a.s.},$$ where $Z = \lim_{N \to \infty}\frac{1}{N}\sum_{n=1}^{N}Y_n$.
I suspect some things. I tried to use the Strong Law of Large Number but nothing... Also since the r.v. are bounded I tried to use Fatou theorem and the Dominated Convergence Theorem but again I could not prove the statement. Any idea?