Consider a probability space with infinitely many coin flips $X_1, X_2, X_3, \dots$. Consider the sequence of $\sigma$-fields $\{\mathcal{F_n}\}$ generated by these random variables. Describe the elements of $\mathcal{F}_1$, $\mathcal{F_2}$, and $\mathcal{F_3}$.
My intial attempt for $\mathcal{F}_1$: $$\mathcal{F}_1 = \big\{ \emptyset, \Omega, \{ (a_1,a_2, \dots): X_1 = 0 \}, \{ (a_1,a_2, \dots): X_1 = 1 \} \big\}.$$
I'm not sure if this is correct, and I'm not quite sure how to proceed with $\mathcal{F_2}$ and $\mathcal{F_3}$. I'm having a difficult time thinking about $\sigma$-fields generated by random variables.
I guess that $\mathscr F_1$ is the $\sigma $-algebra generated by $X_1$. So it is
$$\{\{H,T\}, \{H\}, \{T\},\emptyset\}.$$
Then $\mathscr F_2$ is the $\sigma $-algebra generated by $X_1,X_2$. That is, $$\mathscr F_2=\mathscr F_1\times\mathscr F_1$$
containing $\Omega_2$, the set of the possible pairs of $H $ and $T $, all the outcomes of double coin flipping, and all the subsets of this set:
$$\mathscr F_2=\{\Omega_2\}\cup 2^{\Omega_2}.$$
And for the $\sigma $-algebra generated by $X_1,X_2, X_3$ you will have to do the same with $\Omega_3$, the set of the possible triplets of $H$ and $T$.