Let $\{X_{n,i} : n \in \mathbb{N},i \in I\}$ be real-valued random variables on $(\Omega,\mathcal{F},\mathcal{P})$. Assume that for each $n \in \mathbb{N}$, the variables $\{X_{n,i} : i \in I\}$ are independent. Assume that for each $i \in I$ there is a real-valued random variable $Y_i$ on $(\Omega,\mathcal{F},\mathcal{P})$ such that $X_{n,i} \to Y_i$ a.s. as $n \to \infty$. How can I show that $\{Y_i$ : i $\in I\}$ are independent?
I tried to apply integral convergence theorem to random variables of the type $f(X_{n,i})$ where $f$ is a bounded, continuous function but I was stuck half way.
Hint: this problem is asking if independence of variables is preserved in the limit. To simplify, try to prove that two sequences $\{ X^1_i\},\{X^2_j\}$ of all independent random variables limit to two variables that stay independent.