Independence of $\sigma$-algebra generated by random variables

Question

I found this question (Does $\mathbb{P}$-a.s. convergence preserve independence?).

There is a sequence of R.V. $\{\xi_n\}$ and $\sigma$-algebra $F$. All $\xi_n$ are independent from $F$.

I'm trying to understand answer by martini. It is not clear why $\sigma(X_n: n \in N)$ is independent from $F$ if each $\sigma(X_n)$ is.

Answer 1

If $X$ and $Y$ are i.i.d $\pm 1$ with probability $\frac 1 2 $ each then $X,Y,XY$ are pairwise independent but not jointly independent.

Now let $F=\sigma (X), X_1=Y$ and $X_n =XY, n=2,3...$. Then $F$ is independent of $\sigma (X_n)$ for each $n$ but it is not independent of $\sigma (X_1,X_2,...)$.